  We report the first experimental evidence of nontrivial thermal behavior of
the simplest mesoscopic system - a superconducting loop. By measuring the
specific heat C of an array of 450,000 noninteracting aluminum loops with very
high accuracy of ~20 fJ/K, we show that the loops go through a periodic
sequence of phase transitions (with period of an integer number of magnetic
flux quanta) as the magnetic flux threading each loop is increased. The
transitions are well described by the Ginzburg-Landau theory and are
accompanied by discontinuities of C of only several thousands of Boltzmann
constants k_B.
