Using the Double Chooz detector, designed to measure the neutrino mixing angle $theta_{13}$, the products of $mu^-$ capture on $^{12}$C, $^{13}$C, $^{14}$N and $^{16}$O have been measured. Over a period of 489.5 days, $2.3times10^6$ stopping cosmic $mu^-$ have been collected, of which $1.8times10^5$ captured on carbon, nitrogen, or oxygen nuclei in the inner detector scintillator or acrylic vessels. The resulting isotopes were tagged using prompt neutron emission (when applicable), the subsequent beta decays, and, in some cases, $beta$-delayed neutrons. The most precise measurement of the rate of $^{12}mathrm C(mu^-, u)^{12}mathrm B$ to date is reported: $6.57^{+0.11}_{-0.21}times10^{3},mathrm s^{-1}$, or $(17.35^{+0.35}_{-0.59})%$ of nuclear captures. By tagging excited states emitting gammas, the ground state transition rate to $^{12}$B has been determined to be $5.68^{+0.14}_{-0.23}times10^3,mathrm s^{-1}$. The heretofore unobserved reactions $^{12}mathrm C(mu^-, ualpha)^{8}mathrm{Li}$, $^{13}mathrm C(mu^-, umathrm nalpha)^{8}mathrm{Li}$, and $^{13}mathrm C(mu^-, umathrm n)^{12}mathrm B$ are measured. Further, a population of $beta$n decays following stopping muons is identified with $5.5sigma$ significance. Statistics limit our ability to identify these decays definitively. Assuming negligible production of $^{8}$He, the reaction $^{13}mathrm C(mu^-, ualpha)^{9}mathrm{Li}$ is found to be present at the $2.7sigma$ level. Limits are set on a variety of other processes.
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