We report Hubble Space Telescope observations that provide evidence for a companion to Proxima Centauri (Gl 551), the closest star to the Sun. Data acquired with the Faint Object Spectrograph (FOS), used as a coronagraphic camera, show excess light that can be interpreted as being from a substellar object within 0.5" of Proxima Cen. Two observations of Proxima Cen separated by 103 days indicate a point source (or a feature) superposed on the wing of the point-spread function in the FOS images. This feature moves relative to the aperture, and on the plane of the sky. Comparisons with other FOS images of stars acquired using the coronagraphic mode reveal no comparable features or evidence that this feature can be explained by any instrumental anomaly. If this feature denotes a companion to Proxima Cen, it has an apparent separation corresponding to ~0.5 AU at Proxima Cen and is ~7 mag fainter than Proxima Cen in the bandpass of the FOS red detector. The small apparent separation could result from a highly eccentric orbit, which could project a close companion. Alternately, the small separation could imply a short (~1 yr) period. Further coronagraphic observations, using the Space Telescope Imaging Spectrograph, the Near-Infrared Camera and Multiobject Spectrometer, or some other instrumentation, are needed to verify this tentative result.