The U.S. Naval Observatory CCD trigonometric parallax program is described in detail, including the instrumentation employed, observing procedures followed, and reduction procedures applied. Astrometric results are presented for 72 stars ranging in apparent brightness from V = 15.16 to 19.58. Photometry (V and V - I on the Kron-Cousins system) is presented for the parallax stars and for all 426 individual reference stars employed in the astrometric solutions. Corrections for differential color refraction, calibrated to the observed V - I colors, have been applied to all astrometric measures. The mean errors in the relative parallaxes range from +/- 0.0005" to +/- 0.0027" with a median value of +/- 0.0010". Seventeen of the 23 stars with V(tan) > 200 km s-1 form a well-delineated sequence of extreme subdwarfs covering 11.5 < M(V) < 14.5 in the M(V) vs V - I diagram. The transformation to the M(bol) vs log T(eff) plane is presented and the results are compared with various model interior computations. Within the. limitations due to the uncertain T(eff) scale for cool dwarfs and subdwarfs, the coolest members of the extreme subdwarf sequence appear to be near the hydrogen-burning minimum mass limit for stars with metallicities of [M/H] almost-equal-to -2.