We present a detailed analysis of the heat capacity of a near-perfect S=1/2 kagome antiferromagnet, zinc paratacamite Zn(x)Cu(4-x)(OH)(6)Cl(2), as a function of stoichiometry x-->1 and for fields of up to 9 T. We obtain the heat capacity intrinsic to the kagome layers by accounting for the weak Cu2+/Zn2+ exchange between the Cu and the Zn sites, which was measured independently for x=1 using neutron diffraction. The evolution of the heat capacity for x=0.8...1 is then related to the hysteresis in the magnetic susceptibility. We conclude that for x>0.8 zinc paratacamite is a spin liquid without a spin gap, in which unpaired spins give rise to a macroscopically degenerate ground state manifold with increasingly glassy dynamics as x is lowered.