We present the first radial velocity curve of the companion star to BW Cir that demonstrates the presence of a black hole in this X-ray transient that recorded outbursts in 1987 and 1997 (and possibly 1971-1972). We identify weak absorption features corresponding to a G0-5 III donor star, strongly veiled by a residual accretion disk that contributes 61%-65% of the total light at 6300 Å. The Doppler motions of these features trace an orbit of P=2.54448 days (or its 1 yr alias of P=2.56358 days) and a velocity semiamplitude K2=279+/-5 km s-1 (or K2=292+/-5 km s-1). Both solutions are equally possible. The mass function implied by the shorter period solution is f(M)=5.75+/-0.30 Msolar, which, when combined with the rotational broadening of the tidally locked companion (Vsini=71+/-4 km s-1), yields a compact object mass of M1sin3i=7.34+/-0.46 Msolar. This is substantially above the mass of a neutron star under any standard equation of state of nuclear matter. The companion star is probably a G subgiant that has evolved off the main sequence in order to fill its Roche lobe. Remarkably, a distance of >=27 kpc is inferred by the companion's luminosity, and this is supported by the large observed systemic velocity (γ=103+/-4 km s-1), which requires such a distance in order to be consistent with the Galactic rotation curve.