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We present preliminary trigonometric parallaxes and proper motions for 22 L dwarfs and 18 T dwarfs measured using the ASTROCAM infrared imager on the US Naval Observatory (USNO) 1.55 m Strand Astrometric Reflector. The results presented here are based on observations obtained between 2000 September and 2002 November; about half of the objects have an observational time baseline of Δt=1.3 yr and half Δt=2.0 yr. Despite these short time baselines, the astrometric quality is sufficient to produce significant new results, especially for the nearer T dwarfs. Seven objects are in common with the USNO optical CCD parallax program for quality control and seven in common with the European Southern Observatory 3.5 m New Technology Telescope parallax program. We compare astrometric quality with both of these programs. Relative to absolute parallax corrections are made by employing Two Micron All Sky Survey and/or Sloan Digital Sky Survey photometry for reference-frame stars. We combine USNO infrared and optical parallaxes with the best available California Institute of Technology (CIT) system photometry to determine M_J_, M_H, and M_Kvalues for 37 L dwarfs between spectral types L0 and L8 and 19 T dwarfs between spectral types T0.5 and T8 and present selected absolute magnitude versus spectral type and color diagrams, based on these results. Luminosities and temperatures are estimated for these objects. Of special interest are the distances of several objects that are at or near the L-T dwarf boundary so that this important transition can be better understood. The previously reported early to mid T dwarf luminosity excess is clearly confirmed and found to be present at J, H, and K. The large number of objects that populate this luminosity-excess region indicate that it cannot be due entirely to selection effects. The T dwarf sequence is extended to M_J~16.9 by 2MASS J041519-0935, which, at d=5.74 pc, is found to be the least luminous [log(L/L_☉)=-5.58] and coldest (T_eff~760 K) brown dwarf known. Combining results from this paper with earlier USNO CCD results we find that, in contrast to the L dwarfs, there are no examples of low-velocity (V_tan<20 km/s) T dwarfs. This is consistent with the T dwarfs in this study being generally older than the L dwarfs. We briefly discuss future directions for the USNO infrared astrometry program.