They call this the "Round Barn".

The architectural centerpiece of the Red Mile is
the Round Barn—actually an octagonal building,
topped by a cupola, that originally housed floral
exhibits but later did serve as a barn. (Still later,
it was the place to go to gamble. A city ordinance
had banned that particular activity inside Lexington,
and the Red Mile property was conveniently just outside
the city limits.) Built in 1880, the Round Barn was
financed in part by reparations paid to Lexington by
the U.S. Congress for damage done by Union troops
during the Civil War.
"http://www.ket.org/kentuckylife/700s/kylife712.html"