(Recasts, updates prices)
By John Geddie
LONDON, Dec 13 (Reuters) - Italy's 10-year borrowing costs fell to a one-month low on Tuesday, as UniCredit announced the country's biggest ever share sale in the latest bid to clean up its banking sector.
The premium over Germany that Italy pays to borrow in bond markets fell to its lowest in over a month, having risen sharply ahead of a Dec. 4 referendum.
The country's largest bank said it would raise 13 billion euros ($13.8 billion) to bolster its balance sheet, while new premier Paolo Gentiloni said his government would be ready to take action to support the country's troubled banking sector.
The fate of Italy's financial sector is intrinsically tied to its government because banks are the biggest investors in the country's bonds.
Gentiloni presented an almost unchanged cabinet on Monday, which analysts said was a welcome response to the political upheaval of a constitutional referendum that unseated his predecessor Matteo Renzi.
"The markets appear to be taking the developments in the banking sector quite positively and the cabinet chosen by Gentiloni has reassured investors," DZ Bank strategist Christian Lenk said.
Italy's new government will face confidence votes in both houses of parliament this week. The small centre-right ALA party that supported Renzi said it might not back the new government, raising doubts over whether Gentiloni will have the numbers in parliament to form a majority.
Italian 10-year government bond yields fell as much as 12 basis points to 1.878 percent on Tuesday, a one-month low. It was on track for its biggest one-day fall since Dec. 2, just before the referendum.
Compared to German equivalents, the premium Italy would pay to borrow 10-year money in debt markets fell to 153 basis points, the lowest in over a month.
German yields fell by a lesser extent - 4 basis points - to 0.36 percent, while most other euro zone yields were 5-8 bps lower on the day.
Italian banking stocks gained 5 percent, hauled up by a rise of almost 15 percent in UniCredit which was set for its biggest one-day gain since 2010.
"The uncertainty of a major government reshuffling has been removed and investors appear to be cheering developments in the banking sector as well," Commerzbank strategist David Schnautz said.
1 = 0.9423 euros Reporting by John Geddie; Editing by Tom Heneghan and John Stonestreet