We report a measurement of the ortho-para transition rate in the p mu p molecule. The experiment was conducted at TRIUMF via the measurement of the time dependence of the 5.2 MeV neutrons from muon capture in liquid hydrogen. The measurement yielded an ortho-para rate Lambda op = (11.1 +/- 1.7 +/-(0.9)(0.6)) x 10(4) s(-1), which is substantially larger than the earlier result of Bardin et al. The result has striking implications for the proton's induced pseudoscalar coupling g(p), changing the value of g(p) obtained from the most precise ordinary muon capture measurement from 10.6 +/- 2.7 to 0.8 +/- 2.8, and from the sole radiative muon capture measurement from 12.2 +/- 1.1 to 10.6 +/- 1.2, bringing the latter result closer to theoretical predictions.