  The rate of nuclear muon capture by the proton has been measured using a new
experimental technique based on a time projection chamber operating in
ultra-clean, deuterium-depleted hydrogen gas at 1 MPa pressure. The capture
rate was obtained from the difference between the measured $\mu^-$
disappearance rate in hydrogen and the world average for the $\mu^+$ decay
rate. The target's low gas density of 1% compared to liquid hydrogen is key to
avoiding uncertainties that arise from the formation of muonic molecules. The
capture rate from the hyperfine singlet ground state of the $\mu p$ atom is
measured to be $\Lambda_S=725.0 \pm 17.4 s^{-1}$, from which the induced
pseudoscalar coupling of the nucleon, $g_P(q^2=-0.88 m_\mu^2)=7.3 \pm 1.1$, is
extracted. This result is consistent with theoretical predictions for $g_P$
that are based on the approximate chiral symmetry of QCD.
