  Using a high-resolution differential technique we have determined the
electronic specific heat coefficient gamma(T) of Ba1-xKxFe2As2 with x=0 to 1.0,
at temperatures (T) from 2K to 380K and in magnetic fields H=0 to 13T. In the
normal state gamma_n(x,T) increases strongly with x at low temperature,
compatible with a mass renormalisation ~12 at x=1, and decreases weakly with x
at high temperature. A superconducting transition is seen in all samples from
x=0.2 to 1, with transition temperatures and condensation energies peaking
sharply at x=0.4. Superconducting coherence lengths xi_{ab}~20{\AA} and
xi_c~3{\AA} are estimated from an analysis of Gaussian fluctuations. For many
dopings we see features in the H and T-dependences of gamma_s(T,H) in the
superconducting state that suggest superconducting gaps in three distinct
bands. A broad "knee" and a sharp mean-field-like peak are typical of two
coupled gaps. However, several samples show a shoulder above the sharp peak
with an abrupt onset at T_{c,s} and a T-dependence
gamma_s(T)\propto\sqrt{1-T/T_{c,s}}. We provide strong evidence that the
shoulder is not due to doping inhomogeneity and suggest it is a distinct gap
with an unconventional T-dependence Delta_s(T)\propto(1-T/T_{c,s})^{0.75} near
T_{c,s}. We estimate band fractions and T=0 gaps from 3-band alpha-model fits
to our data and compare the x-dependences of the band fractions with
spectroscopic studies of the Fermi surface.
