Near-infrared J, H, and K_s photometric measurements of 92 Cepheids in the Large Magellanic Cloud are presented. The stars are spread over the face of the Cloud, their periods range from 3 to 100 days, and their light curves are sampled at an average of 22 phase points per star. The intensity-weighted mean magnitudes and colors define period-luminosity-color (PL or PLC) relations whose uncertainties due to differential metal abundance and reddening/extinction effects are minimal. The dispersions in the infrared PL, PLC, and extinction-free period-Wesenheit relations are extremely small, amounting to less than 0.10 mag (or 5% in distance). The orientation of the disk plane of the sample (inclination angle and line of nodes) agrees well with the 2001 results of van der Marel & Cioni. The PL and PLC fits are the best-determined such relationships yet found for any sample of Cepheids and establish a calibration that can be used to precisely anchor ground- and space-based near-infrared Cepheid data to external galaxies, as well as back to Cepheid calibrators in the Galaxy. As an example, we use the 1998 Galactic Cepheid calibration of Gieren and coworkers to obtain the distance modulus to the centroid of our LMC sample. The true modulus of the LMC is thus determined to be 18.50 ± 0.05 mag. Currently, the dominant source of uncertainty in this number is the scatter in the Galactic calibrator sample. The PLC fits and dispersions and the dependence of the PLC on metal abundance are compared with theoretical versions computed from the 1999 work of Alibert and coworkers. Overall, the agreement is excellent, indicating that at near-infrared wavelengths the slope and dispersion of the PLC depend very weakly on metal abundance. The shift in the JHK PLC relations is ~0.02 mag for a change in metal abundance from solar to one-half solar.