1E 1740.7 - 2942 is thought to be a Cygnus X-1-like black hole near the Galactic center, although sever important questions regarding its nature and possible connection to the narrow e+-e- annihilation line seen from this region remain unanswered. Here we report on HK photometry of this field which shows the distribution of near-infrared stars is consistent with a population lying near the Galactic center similar to that of the late-type Baade's window population with an average foreground extinction A(V) = 30 +/- 5 mag and a substantially greater value of A(V) = 55 +/- 5 mag associated with stars centered at the position of the molecular cloud enshrouding 1E 1740.7-2942. We argue that the known distribution of bulge stars and the observed ''shadowing'' by, the cloud constrain its position to lie within approximately 300 pc of the Galactic center. We show that the implied total hydrogen column density in this direction is (1.05 +/- 0.53) x 10(23) atoms cm-2, of Which (0.48 +/- 0.24) x 10(23) cm-2 is attributable to the cloud itself. This is consistent with the value required to account for the attenuation in the 4-300 keV spectrum of 1E 1740.7-2942 in its standard state, and we therefore conclude that 1E 1740.7-2942 is at least as distant as the molecular cloud. The total mass of the observed obscuring material associated with the cloud is (4.6 +/- 2.3) x 10(3) M, an order of magnitude below that derived under the assumption that the molecular cloud is predominantly composed of gas traced by HCO+ emission with an average number density greater than 5 x 10(4) atoms cm-3. The corresponding average cloud density of approximately 5 x 10(3) cm-3 appears to be inconsistent with the value required to account for the narrow electron-positron line flux variations if the leptons trace out straight-line trajectories.