We are undertaking an astrometric search for gas giant planets and brown dwarfs orbiting nearby low mass dwarf stars with the 2.5-m du Pont telescope at the Las Campanas Observatory in Chile. We have built two specialized astrometric cameras, the Carnegie Astrometric Planet Search Cameras (CAPSCam-S and CAPSCam-N), using two Teledyne Hawaii-2RG HyViSI arrays, with the cameras' design having been optimized for high accuracy astrometry of M dwarf stars. We describe two independent CAPSCam data reduction approaches and present a detailed analysis of the observations to date of one of our target stars, NLTT 48256. Observations of NLTT 48256 taken since July 2007 with CAPSCam-S imply that astrometric accuracies of around 0.3 milliarcsec per hour are achievable, sufficient to detect a Jupiter-mass companion orbiting 1 AU from a late M dwarf 10 pc away with a signal-to-noise ratio of about 4. We plan to follow about 100 nearby (primarily within about 10 pc) low mass stars, principally late M, L, and T dwarfs, for 10 years or more, in order to detect very low mass companions with orbital periods long enough to permit the existence of habitable, Earth-like planets on shorter-period orbits. These stars are generally too faint and red to be included in ground-based Doppler planet surveys, which are often optimized for FGK dwarfs. The smaller masses of late M dwarfs also yield correspondingly larger astrometric signals for a given mass planet. Our search will help to determine whether gas giant planets form primarily by core accretion or by disk instability around late M dwarf stars.