Mayor Bloomberg and the Ballyshear estateMayor Michael Bloomberg has signed a contract to purchase a $20 million Hamptons estate overlooking Peconic Bay, sources told the Post. Those sources also said that there were pages upon pages worth of confidentiality agreements involved in the deal, but apparently that didn’t help to quiet the rumor mill, as it seems the cat’s out of the bag on Bloomie’s future home at 119 Whites Lane. The historic, 35-acre Shinnecock Hills property, called Ballyshear, borders the posh National Golf Links of America golf course and has the touches of several noted architects. The 22,000-square-foot Georgian mansion has 11 bedrooms and eight-and-a-half bathrooms. It was built in 1910 by F. Burrall Hoffman and is surrounded by English gardens designed by Frederick Law Olmstead and Rose Standish Nichols, according to the listing by Paul Brennan and Ronald White of Prudential Douglas Elliman, which was last priced at $22.5 million. Other amenities include a half-mile, tree-lined driveway leading up to the main house, as well as a pool, tennis court, horse stables and trails. Property taxes total $42,110 per year. [Post]