  We report multiple lines of evidence for a stochastic signal that is
correlated among 67 pulsars from the 15-year pulsar-timing data set collected
by the North American Nanohertz Observatory for Gravitational Waves. The
correlations follow the Hellings-Downs pattern expected for a stochastic
gravitational-wave background. The presence of such a gravitational-wave
background with a power-law-spectrum is favored over a model with only
independent pulsar noises with a Bayes factor in excess of $10^{14}$, and this
same model is favored over an uncorrelated common power-law-spectrum model with
Bayes factors of 200-1000, depending on spectral modeling choices. We have
built a statistical background distribution for these latter Bayes factors
using a method that removes inter-pulsar correlations from our data set,
finding $p = 10^{-3}$ (approx. $3\sigma$) for the observed Bayes factors in the
null no-correlation scenario. A frequentist test statistic built directly as a
weighted sum of inter-pulsar correlations yields $p = 5 \times 10^{-5} - 1.9
\times 10^{-4}$ (approx. $3.5 - 4\sigma$). Assuming a fiducial $f^{-2/3}$
characteristic-strain spectrum, as appropriate for an ensemble of binary
supermassive black-hole inspirals, the strain amplitude is $2.4^{+0.7}_{-0.6}
\times 10^{-15}$ (median + 90% credible interval) at a reference frequency of
1/(1 yr). The inferred gravitational-wave background amplitude and spectrum are
consistent with astrophysical expectations for a signal from a population of
supermassive black-hole binaries, although more exotic cosmological and
astrophysical sources cannot be excluded. The observation of Hellings-Downs
correlations points to the gravitational-wave origin of this signal.
