We determine the mass of the nuclear black hole ($M$) in NGC 3706, an early type galaxy with a central surface brightness minimum arising from an apparent stellar ring, which is misaligned with respect to the galaxys major axis at larger radii. We fit new HST/STIS and archival data with axisymmetric orbit models to determine $M$, mass-to-light ratio ($Upsilon_V$), and dark matter halo profile. The best-fit model parameters with 1$sigma$ uncertainties are $M = (6.0^{+0.7}_{-0.9}) times 10^8 M_{scriptscriptstyle odot}$ and $Upsilon_V = 6.0 pm 0.2 M_{scriptscriptstyle odot} L_{{scriptscriptstyle odot},V}^{-1}$ at an assumed distance of 46 Mpc. The models are inconsistent with no black hole at a significance of $Deltachi^2 = 15.4$ and require a dark matter halo to adequately fit the kinematic data, but the fits are consistent with a large range of plausible dark matter halo parameters. The ring is inconsistent with a population of co-rotating stars on circular orbits, which would produce a narrow line-of-sight velocity distribution (LOSVD). Instead, the rings LOSVD has a small value of $|V|/sigma$, the ratio of mean velocity to velocity dispersion. Based on the observed low $|V|/sigma$, our orbit modeling, and a kinematic decomposition of the ring from the bulge, we conclude that the stellar ring contains stars that orbit in both directions. We consider potential origins for this unique feature, including multiple tidal disruptions of stellar clusters, a change in the gravitational potential from triaxial to axisymmetric, resonant capture and inclining of orbits by a binary black hole, and multiple mergers leading to gas being funneled to the center of the galaxy.
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