We report 850 μm dust polarization observations of a low-mass (∼12 M o) starless core in the ρ 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 survey. We detect an ordered magnetic field projected on the plane of the sky in the starless core. The magnetic field across the ∼0.1 pc core shows a predominant northeast-southwest orientation centering between ∼40° and ∼100°, 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 increasing total intensity (I), with a power-law index of -1.03 ± 0.05. We estimate the plane-of-sky field strength (B pos) using modified Davis-Chandrasekhar-Fermi methods based on structure function (SF), autocorrelation function (ACF), and unsharp masking (UM) analyses. We find that the estimates from the SF, ACF, and UM methods yield strengths of 103 ± 46 μG, 136 ± 69 μG, and 213 ± 115 μG, 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.