  Pulsar timing array collaborations, such as the North American Nanohertz
Observatory for Gravitational Waves (NANOGrav), are seeking to detect nanohertz
gravitational waves emitted by supermassive black hole binaries formed in the
aftermath of galaxy mergers. We have searched for continuous waves from
individual circular supermassive black hole binaries using the NANOGrav's
recent 12.5-year data set. We created new methods to accurately model the
uncertainties on pulsar distances in our analysis, and we implemented new
techniques to account for a common red noise process in pulsar timing array
data sets while searching for deterministic gravitational wave signals,
including continuous waves. As we found no evidence for continuous waves in our
data, we placed 95\% upper limits on the strain amplitude of continuous waves
emitted by these sources. At our most sensitive frequency of 7.65 nanohertz, we
placed a sky-averaged limit of $h_0 < $ $(6.82 \pm 0.35) \times 10^{-15}$, and
$h_0 <$ $(2.66 \pm 0.15) \times 10^{-15}$ in our most sensitive sky location.
Finally, we placed a multi-messenger limit of $\mathcal{M} <$ $(1.41 \pm 0.02)
\times 10^9 M_\odot$ on the chirp mass of the supermassive black hole binary
candidate 3C~66B.
