  We search for an isotropic stochastic gravitational-wave background (GWB) in
the $12.5$-year pulsar timing data set collected by the North American
Nanohertz Observatory for Gravitational Waves. Our analysis finds strong
evidence of a stochastic process, modeled as a power-law, with common amplitude
and spectral slope across pulsars. The Bayesian posterior of the amplitude for
an $f^{-2/3}$ power-law spectrum, expressed as the characteristic GW strain,
has median $1.92 \times 10^{-15}$ and $5\%$--$95\%$ quantiles of $1.37$--$2.67
\times 10^{-15}$ at a reference frequency of $f_\mathrm{yr} = 1
~\mathrm{yr}^{-1}$. The Bayes factor in favor of the common-spectrum process
versus independent red-noise processes in each pulsar exceeds $10,000$.
However, we find no statistically significant evidence that this process has
quadrupolar spatial correlations, which we would consider necessary to claim a
GWB detection consistent with general relativity. We find that the process has
neither monopolar nor dipolar correlations, which may arise from, for example,
reference clock or solar system ephemeris systematics, respectively. The
amplitude posterior has significant support above previously reported upper
limits; we explain this in terms of the Bayesian priors assumed for intrinsic
pulsar red noise. We examine potential implications for the supermassive black
hole binary population under the hypothesis that the signal is indeed
astrophysical in nature.

High Energy Astrophysical Phenomena;Astrophysics of Galaxies;General Relativity and Quantum Cosmology