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We have measured near-infrared magnitudes in the J and K bands for 56 Cepheid variables in the Local Group galaxy NGC 6822 with well-determined periods and optical light curves in the V and I bands. Using the template light-curve approach of Soszy{nacute}ski and coworkers, accurate mean magnitudes were obtained from these data, which allowed us to determine with unprecedented accuracy the distance to NGC 6822 from a multiwavelength period-luminosity solution in the VIJK bands. From our data, we obtain a distance to NGC 6822 of (m-M)₀=23.312±0.021 (random error) mag, with an additional systematic uncertainty of ∼3%. This distance value is tied to an assumed LMC distance modulus of 18.50. From our multiwavelength approach, we find for the total (average) reddening to the NGC 6822 Cepheids E(B-V)=0.356±0.013 mag, which is in excellent agreement with a previous determination by McGonegal and coworkers from near-infrared photometry and implies significant internal reddening of the Cepheids in NGC 6822. Our present, definitive distance determination of NGC 6822 from Cepheids agrees within 2% with the previous distance we had derived from optical photometry alone, but has significantly reduced error bars. Our Cepheid distance to NGC 6822 is in excellent agreement with the recent independent determination of Cioni & Habing from the I-band magnitude of the tip of the red giant branch. It also agrees well, within the errors, with the early determination of McGonegal et al. ((ref???)1983) from random-phase H-band photometry of nine Cepheids.