We present parallaxes, proper motions, and $J$-band photometry for 348 L and T dwarfs measured using the wide-field near-infrared camera WFCAM on the United Kingdom Infrared Telescope. This is the largest single batch of infrared parallaxes for brown dwarfs to date. Our parallaxes have a median uncertainty of 3.5 mas, similar to most previous ground-based infrared parallax surveys. Our target list was designed to complete a volume-limited parallax sample of L0-T8 dwarfs out to 25 pc spanning declinations $-30^circ$ to $+60^circ$ (68% of the sky). We report the first parallaxes for 165 objects, and we improve on previous measurements for another 53 objects. Our targets include 104 objects (mostly early-L dwarfs) having $Gaia$ DR2 parallax measurements, with which our parallaxes are consistent. We include an extensive comparison of previous literature parallaxes for L and T dwarfs with both our results and $Gaia$ DR2 measurements, identifying systematic offsets for some previous surveys. Our parallaxes confirm that 14 objects previously identified as wide common proper motion companions to main-sequence stars have distances consistent with companionship. We also report new $J_mathrm{MKO}$ photometry for our targets, including the first measurements for 193 of our targets and improvements over previously published $J_mathrm{MKO}$ photometry for another 60 targets. Altogether, our parallaxes will enable the first population studies using a volume-limited sample spanning spectral types L0-T8 defined entirely by parallaxes.
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