Exhibit 99.1

FOR IMMEDIATE RELEASE

Contact: Dana C. Gavenda

President and Chief Executive Officer

FSB Community Bankshares, Inc.

FSB Bancorp, Inc.

Tel. (585) 223-9080

FSB COMMUNITY BANKSHARES, INC. ANNOUNCES RESULTS OF

SPECIAL MEETINGS OF STOCKHOLDERS AND MEMBERS

AND PRELIMINARY OFFERING RESULTS

Fairport, New York, June 29, 2016 — FSB Community Bankshares, Inc. (the “Company”) (OTC Pink: FSBC), the holding company for Fairport Savings Bank, announced that, at separate meetings held today, both the stockholders of the Company and the members of FSB Community Bankshares, MHC, the Company’s parent mutual holding company, approved the Plan of Conversion and Reorganization pursuant to which the FSB Community Bankshares, MHC will convert from the mutual holding company to the stock holding company form of organization.

The Company also reported that the subscription and community offering of the common stock of FSB Bancorp, Inc., the proposed new holding company for Fairport Savings Bank, being undertaken as part of the conversion concluded on June 14, 2016 and was oversubscribed.

The completion of the conversion and offering remain subject to customary regulatory approvals and the satisfaction of customary closing conditions. Subject to receipt of those approvals and satisfaction of those conditions, the Company anticipates closing the transaction in mid-July. Final results of the offering and the anticipated closing date will be announced following receipt of all regulatory approvals to complete the transaction.

The shares of the Company will continue to trade on the OTC Pink Marketplace under the trading symbol “FSBC” through the closing date of the conversion. Following closing, it is anticipated that the shares of FSB Bancorp common stock will trade on the Nasdaq Capital Market under the same trading symbol.

Fairport Savings Bank, founded in 1888, is headquartered in Fairport, New York with four additional full-service branch offices located in Penfield, Irondequoit, Webster and Perinton, New York. The Bank also has loan production offices located in Buffalo, Greece, Pittsford and Watertown, New York. At March 31, 2016, the Company had total assets of $257.8 million.

Certain statements contained herein are “forward-looking statements” within the meaning of Section 27A of the Securities Act of 1933 and Section 21E of the Securities Exchange Act of 1934. Such forward-looking statements may be identified by reference to a future period or periods, or by the use of forward-looking terminology, such as “may,” “will,” “believe,” “expect,” “estimate,” “anticipate,” “continue,” or similar terms or variations on those terms, or

the negative of those terms. Forward-looking statements are subject to numerous risks and uncertainties, including, but not limited to: general economic trends, changes in interest rates, increased competition, changes in consumer demand for financial services, fiscal and monetary policies of the U.S. Government, and changes in government regulations affecting financial institutions, including regulatory compliance costs and capital requirements.

Readers are cautioned not to place undue reliance on any such forward-looking statements, which speak only as of the date made. The factors listed above could affect FSB Bancorp’s and the Company’s financial performance and could cause FSB Bancorp’s and the Company’s actual results for future periods to differ materially from any opinions or statements expressed with respect to future periods in any current statements. FSB Bancorp and the Company do not undertake and specifically decline any obligation to publicly release the results of any revisions, which may be made to any forward-looking statements to reflect events or circumstances after the date of such statements or to reflect the occurrence of anticipated or unanticipated events, except as required to be reported under the rules and regulations of the United States Securities and Exchange Commission.

A registration statement relating to FSB Bancorp’s common stock has been filed with the United States Securities and Exchange Commission. This press release is neither an offer to sell nor a solicitation of an offer to buy FSB Bancorp common stock. The offer is made only by means of the written prospectus forming part of the registration statement (and, in the case of the subscription and community offering, an accompanying stock order form).

The shares of common stock of FSB Bancorp are not savings accounts or savings deposits, may lose value and are not insured by the Federal Deposit Insurance Corporation or any other government agency.