We report 850~$\mu$m dust polarization observations of a low-mass ($\sim$12
$M_{\odot}$) starless core in the $\rho$ Ophiuchus cloud, Ophiuchus C, made
with the POL-2 instrument on the James Clerk Maxwell Telescope (JCMT) as part
of the JCMT B-fields In STar-forming Region Observations (BISTRO) survey. We
detect an ordered magnetic field projected on the plane of sky in the starless
core. The magnetic field across the $\sim$0.1~pc core shows a predominant
northeast-southwest orientation centering between $\sim$40$^\circ$ to
$\sim$100$^\circ$, indicating that the field in the core is well aligned with
the magnetic field in lower-density regions of the cloud probed by
near-infrared observations and also the cloud-scale magnetic field traced by
Planck observations. The polarization percentage ($P$) decreases with an
increasing total intensity ($I$) with a power-law index of $-$1.03 $\pm$ 0.05.
We estimate the plane-of-sky field strength ($B_{\mathrm{pos } }$) using modified
Davis-Chandrasekhar-Fermi (DCF) methods based on structure function (SF),
auto-correlation (ACF), and unsharp masking (UM) analyses. We find that the
estimates from the SF, ACF, and UM methods yield strengths of 103 $\pm$ 46
$\mu$G, 136 $\pm$ 69 $\mu$G, and 213 $\pm$ 115 $\mu$G, respectively. Our
calculations suggest that the Ophiuchus C core is near magnetically critical or
slightly magnetically supercritical (i.e. unstable to collapse). The total
magnetic energy calculated from the SF method is comparable to the turbulent
energy in Ophiuchus C, while the ACF method and the UM method only set upper
limits for the total magnetic energy because of large uncertainties.