  We present an investigation of magnetic ordering in the two-dimensional S=1/2
quantum magnet Cu(Pz)_2(ClO_4)_2 using specific heat and zero field muon-spin
relaxation (\mu^+SR). The magnetic contribution to the specific heat is
consistent with an exchange strength of 17.7(3) K. We find unambiguous evidence
for a transition to a state of three-dimensional long range order below a
critical temperature T_N=4.21(1) K using \mu^+SR even though there is no
feature in the specific heat at that temperature. The absence of a specific
heat anomaly at T_N is consistent with recent theoretical predictions. The
ratio of T_N/J=0.24 corresponds to a ratio of intralayer to interlayer exchange
constants of |J'/J|=6.8x10^-4, indicative of excellent two-dimensional
isolation. The scaled magnetic specific heat of [Cu(Pz)_2(HF_2)]BF_4, a
compound with an analogous structure, is essentially identical to that of
Cu(Pz)_2(ClO_4)_2 although both differ slightly from the predicted value for an
ideal 2D S=1/2 Heisenberg antiferromagnet.
