  We demonstrate a robust experimental method for determining the depth of
individual shallow Nitrogen-Vacancy (NV) centers in diamond with $\sim1$ nm
uncertainty. We use a confocal microscope to observe single NV centers and
detect the proton nuclear magnetic resonance (NMR) signal produced by objective
immersion oil, which has well understood nuclear spin properties, on the
diamond surface. We determine the NV center depth by analyzing the NV NMR data
using a model that describes the interaction of a single NV center with the
statistically-polarized proton spin bath. We repeat this procedure for a large
number of individual, shallow NV centers and compare the resulting NV depths to
the mean value expected from simulations of the ion implantation process used
to create the NV centers, with reasonable agreement.
