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First Trust Energy Infrastructure Fund Issues Notice Regarding June 2018 Distribution

The Board of Trustees of First Trust Energy Infrastructure Fund (the
“Fund”) (NYSE: FIF), CUSIP 33738C103, previously approved a managed
distribution policy for the Fund (the “Managed Distribution Plan”) in
reliance on exemptive relief received from the Securities and Exchange
Commission which permits the Fund to make periodic distributions of
long-term capital gains as frequently as monthly each tax year.

The Fund has declared a distribution payable on June 15, 2018, to
shareholders of record as of June 4, 2018, with an ex-dividend date of
June 1, 2018. This Notice is meant to provide you information about the
sources of your Fund’s distributions. You should not draw any
conclusions about the Fund’s investment performance from the amount of
its distribution or from the terms of its Managed Distribution Plan.

The following tables set forth the estimated amounts of the current
distribution and the cumulative distributions paid this fiscal year to
date for the Fund from the following sources: net investment income
(“NII”); net realized short-term capital gains (“STCG”); net realized
long-term capital gains (“LTCG”); and return of capital (“ROC”). These
estimates are based upon information as of May 31, 2018, are calculated
based on a generally accepted accounting principles (“GAAP”) basis and
include the prior fiscal year-end undistributed net investment income.
The amounts and sources of distributions are expressed per common share.

Ticker

Cusip

Year End

Distribution

NII

STCG

LTCG

ROC (2)

NII

STCG

LTCG

ROC(2)

% of NAV(3)

on NAV(4)

Total

Distributions as

Ticker

Cusip

Year End

Distributions(1)

NII

STCG

LTCG

ROC (2)

NII

STCG

LTCG

ROC(2)

a % of NAV(3)

on NAV(4)

(1) Includes the most recent monthly distribution paid on June 15, 2018.

(2) The Fund estimates that it has distributed more than its income and
net realized capital gains; therefore, a portion of your distribution
may be a return of capital. A return of capital may occur, for example,
when some or all of the money that you invested in the Fund is paid back
to you. A return of capital distribution does not necessarily reflect
the Fund’s investment performance and should not be confused with
“yield” or “income”.

(3) Based on Net Asset Value (“NAV”) as of May 31, 2018.

(4) Total Returns are through May 31, 2018.

(5) The Fund anticipates that, due to the tax treatment of cash
distributions made by Master Limited Partnerships in which the Fund
invests, a portion of distributions the Fund makes to Common
Shareholders may consist of a tax-deferred return of capital.

The amounts and sources of distributions reported in this Notice are
only estimates and are not being provided for tax reporting purposes.
The actual amounts and sources of the amounts for tax reporting purposes
will depend upon the Fund’s investment experience during the remainder
of its fiscal year and may be subject to changes based on tax
regulations. The Fund will send you a Form 1099-DIV for the calendar
year that will tell you how to report these distributions for federal
income tax purposes. You should not use this Notice as a substitute for
your Form 1099-DIV.

First Trust Advisors L.P. (“FTA”), a federally registered investment
advisor, and its affiliate First Trust Portfolios L.P. (“FTP”), a FINRA
registered broker-dealer, are privately held companies that provide a
variety of investment services. FTA is the investment advisor to
exchange-traded funds, closed-end funds, mutual funds, separate managed
accounts and provides supervisory services to FTP sponsored unit
investment trusts. FTA’s assets under management were approximately $126
billion as of May 31, 2018. This includes the supervisory services FTA
provides to FTP sponsored unit investment trusts, which are unmanaged.
FTP is a sponsor of unit investment trusts and distributor of mutual
fund shares and exchange-traded fund creation units. FTA and FTP are
based in Wheaton, Illinois.

Energy Income Partners, LLC (“EIP”) serves as the Fund’s investment
sub-advisor and provides advisory services to a number of investment
companies and partnerships for the purpose of investing in MLPs and
other energy infrastructure securities. EIP is one of the early
investment advisors specializing in this area. As of May 31, 2018, EIP
managed or supervised approximately $6.1 billion in client assets.

Past performance is no assurance of future results. Investment return
and market value of an investment in the Fund will fluctuate. Shares,
when sold, may be worth more or less than their original cost.

Principal Risk Factors: The Fund is subject to risks, including the fact
that it is a non-diversified closed-end management investment company.

Because the Fund is concentrated in securities issued by energy
infrastructure companies, it will be more susceptible to adverse
economic or regulatory occurrences affecting that industry, including
high interest costs, high leverage costs, the effects of economic
slowdown, surplus capacity, increased competition, uncertainties
concerning the availability of fuel at reasonable prices, the effects of
energy conservation policies and other factors.

The Fund invests in securities of non-U.S. issuers which are subject to
higher volatility than securities of U.S. issuers. Because the Fund
invests in non-U.S. securities, you may lose money if the local currency
of a non-U.S. market depreciates against the U.S. dollar.

There can be no assurance as to what portion of the distributions paid
to the Fund’s Common Shareholders will consist of tax-advantaged
qualified dividend income.

Use of leverage can result in additional risk and cost, and can magnify
the effect of any losses.

The information presented is not intended to constitute an investment
recommendation for, or advice to, any specific person. By providing this
information, First Trust is not undertaking to give advice in any
fiduciary capacity within the meaning of ERISA and the Internal Revenue
Code. First Trust has no knowledge of and has not been provided any
information regarding any investor. Financial advisors must determine
whether particular investments are appropriate for their clients. First
Trust believes the financial advisor is a fiduciary, is capable of
evaluating investment risks independently and is responsible for
exercising independent judgment with respect to its retirement plan
clients.

The risks of investing in the Fund are spelled out in the shareholder
reports and other regulatory filings.

Forward-Looking Statements

Certain statements made in this press release that are not historical
facts are referred to as “forward-looking statements” under the U.S.
federal securities laws. Actual future results or occurrences may differ
significantly from those anticipated in any forward-looking statements
due to numerous factors. Generally, the words “believe,” “expect,”
“intend,” “estimate,” “anticipate,” “project,” “will” and similar
expressions identify forward-looking statements, which generally are not
historical in nature. Forward-looking statements are subject to certain
risks and uncertainties that could cause actual results to differ from
those anticipated in any forward-looking statements. You should not
place undue reliance on forward-looking statements, which speak only as
of the date they are made. The Fund undertakes no responsibility to
update publicly or revise any forward-looking statements.

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