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We present new near-infrared (NIR), JHK_s, time-series observations of RR Lyrae variables in the Messier 3 (NGC 5272) globular cluster using the WIRCam instrument at the 3.6 m Canada-France-Hawaii Telescope. Our observations cover a sky area of ∼21' x 21' around the cluster center and provide an average of 20 epochs of homogeneous JHK_s-band photometry. New homogeneous photometry is used to estimate robust mean magnitudes for 175 fundamental-mode (RRab), 47 overtone-mode (RRc), and 11 mixed-mode (RRd) variables. Our sample of 233 RR Lyrae variables is the largest thus far obtained in a single cluster with time-resolved, multiband NIR photometry. NIR-to-optical amplitude ratios for RR Lyrae in Messier 3 exhibit a systematic increase moving from RRc to short-period (P < 0.6 day) and long-period (P >= 0.6 day) RRab variables. We derive JHK_s-band period-luminosity relations for RRab, RRc, and the combined sample of variables. Absolute calibrations based on the theoretically predicted period-luminosity-metallicity relations for RR Lyrae stars yield a distance modulus, µ=15.041±0.017(statistical)±0.036(systematic) mag, to Messier 3. When anchored to trigonometric parallaxes for nearby RR Lyrae stars from the Hubble Space Telescope and the Gaia mission, our distance estimates are consistent with those resulting from the theoretical calibrations, albeit with relatively larger systematic uncertainties.