We have obtained exposures of the field of X0512-401 in the globular cluster NGC 1851, in X-rays with the Chandra X-Ray Observatory, and in the far-UV with the Hubble Space Telescope. We derive an accurate new X-ray position (within ~1'') for X0512-401, which enables us to confirm that the only plausible candidate for the optical/UV counterpart is star A, which we previously identified from Wide Field Planetary Camera 2 imaging. We find no evidence for X-ray or UV flux modulation on the ultrashort (1 hr) expected binary period, which implies a low system inclination. In addition, we have detected and spatially resolved an X-ray burst event, confirming the association of the burster, quiescent X-ray source, and optical object. The very large LX/Lopt of this object implies an extraordinarily compact system, similar to the sources in NGC 6624 and NGC 6712.