ATHENS (Reuters) -Greece's state-controlled bank bailout fund HFSF said on Monday it sold its 9% stake in Alpha Bank to UniCredit for 293.5 million euros ($314 million) after an improved bid by the Italian bank.

UniCredit announced last month it would become Alpha Bank's biggest investor by buying a 9% stake owned by Greece's HFSF. As part of the deal it also agreed to acquire most of Alpha's Romanian business for 300 million euros.

The fund said UniCredit had bought its Alpha shares at a price of 1.39 euros each, representing a premium of 9.4% to the stock's closing price on the last trading day before the initial announcement.

Shares in Alpha closed at 1.27 euros each on Oct. 20, the Friday before UniCredit unveiled the deal, which also includes a commercial partnership to distribute financial products to Alpha's customers.

Under Chief Executive Andrea Orcel, the former head of investment banking at Swiss bank UBS, UniCredit has been working to rebuild fee-yielding businesses which it had depleted during its crisis years to beef up capital.

Greece has been attracting significant investment as its economy strengthened after the crisis, prompting S&P Global to upgrade the country to investment grade in October.

HFSF sold last month its 1.4% stake in Eurobank and is expected to proceed with the sale of about 20% of National Bank in the coming days.

It acquired stakes in the four biggest lenders in the previous decade, injecting about 50 billion euros to recapitalise them during the country's financial crisis.

Lazard Frères acted as disposal adviser to the fund and Rothschild & Co as divestment strategy adviser.

The UK arm of U.S. law firm Skadden, Arps, Slate, Meagher & Flom and Greece's Kyriakides Georgopoulos acted as external legal counsel to HFSF. ($1 = 0.9355 euros)

(Reporting by Lefteris Papadimas; Editing by Tom Hogue, Stephen Coates and Keith Weir)