  The first measurement of the elementary process $\mu^- p \rightarrow
\nu_{\mu} n \gamma$ is reported. A photon pair spectrometer was used to measure
the partial branching ratio ($2.10 \pm 0.22) \times 10^{-8}$ for photons of k >
60 MeV. The value of the weak pseudoscalar coupling constant determined from
the partial branching ratio is $g_p(q^{2}=-0.88m_{\mu}^2) = (9.8 \pm 0.7 \pm
0.3) \cdot g_a(0)$, where the first error is the quadrature sum of statistical
and systematic uncertainties and the second error is due to the uncertainty in
$\lambda_{op}$, the decay rate of the ortho to para $p \mu p$ molecule. This
value of g_p is $\sim$1.5 times the prediction of PCAC and pion-pole dominance.
