  We have obtained deep images in the near-infrared J and K filters for several
fields in the Local Group galaxy WLM. We report intensity mean magnitudes for
31 Cepheids located in these fields which we previously discovered in a
wide-field optical imaging survey of WLM. The data define tight
period-luminosity relations in both near-infrared bands which we use to derive
the total reddening of the Cepheids in WLM and the true distance modulus of the
galaxy from a multiwavelength analysis of the reddened distance moduli in the
VIJK bands. From this, we obtain the values E(B-V) = 0.082 $\pm$ 0.02, and
$(m-M)_{0} = 24.924 \pm 0.042$ mag, with a systematic uncertainty in the
distance of about $\pm$ 3%. This Cepheid distance agrees extremely well with
the distance of WLM determined from the I-band TRGB method by ourselves and
others. Most of the reddening of the Cepheids in WLM (0.06 mag) is produced
inside the galaxy, demonstrating again the need for an accurate determination
of the total reddening and/or the use of infrared photometry to derive Cepheid
distances which are accurate to 3% or better, even for small irregular galaxies
like WLM.
