  We report the results of an infrared Doppler survey designed to detect brown
dwarf and giant planetary companions to a magnitude-limited sample of ultracool
dwarfs. Using the NIRSPEC spectrograph on the Keck II telescope, we obtained
approximately 600 radial velocity measurements over a period of six years for a
sample of 59 late-M and L dwarfs spanning spectral types M8/L0 to L6. A
subsample of 46 of our targets have been observed on three or more epochs. We
rely on telluric CH4 absorption features in the Earth's atmosphere as a
simultaneous wavelength reference and exploit the rich set of CO absorption
features found in the K-band spectra of cool stars and brown dwarfs to measure
radial velocities and projected rotational velocities. For a bright, slowly
rotating M dwarf standard we demonstrate a radial velocity precision of 50 m/s,
and for slowly rotating L dwarfs we achieve a typical radial velocity precision
of approximately 200 m/s. This precision is sufficient for the detection of
close-in giant planetary companions to mid-L dwarfs as well as more equal mass
spectroscopic binary systems with small separations (a<2 AU). We present an
orbital solution for the subdwarf binary LSR1610-0040 as well as an improved
solution for the M/T binary 2M0320-04. We also combine our radial velocity
measurements with distance estimates and proper motions from the literature to
estimate the dispersion of the space velocities of the objects in our sample.
Using a kinematic age estimate we conclude that our UCDs have an age of
5.0+0.7-0.6 Gyr, similar to that of nearby sun-like stars. We simulate the
efficiency with which we detect spectroscopic binaries and find that the rate
of tight (a<1 AU) binaries in our sample is 2.5+8.6-1.6%, consistent with
recent estimates in the literature of a tight binary fraction of 3-4%.
(abridged)
