  We have conducted an extensive wide-field imaging survey for Cepheid
variables in the Local Group irregular galaxy WLM. From data obtained on 101
nights, we have discovered 60 Cepheids which include 14 of the 15 Cepheid
variables previously detected by Sandage and Carlson. Down to a period of 3
days, our Cepheid survey in WLM should be practically complete. Importantly, we
have found for the first time a long-period Cepheid (P=54.2 days) in this
galaxy, alleviating the puzzle that WLM with its many blue, massive stars does
not contain Cepheids with periods longer than about 10 days. Our data define
tight period-luminosity relations in V, I and the reddening-free Wesenheit
magnitude ${\rm W}_{\rm I}$ which are all extremely well fit by the
corresponding slopes of the LMC Cepheid PL relation, suggesting no change of
the PL relation slope down to a Cepheid metal abundance of about -1.0 dex, in
agreement with other recent studies. We derive a true distance modulus to WLM
of 25.144 $\pm$0.03 (r) $\pm$0.07 (s) mag from our data, in good agreement with
the earlier 24.92 $\pm$ 0.21 mag determination of Lee, Freedman and Madore
(1993a) from Cepheid variables. The quoted value of the systematic uncertainty
does not include the contribution from the LMC distance which we have assumed
to be 18.50 mag, as in the previous papers in our project.
