The repetition threshold is a measure of the extent to which
there need to be consecutive (partial) repetitions of finite
words within infinite words
over alphabets of various sizes. Dejean's Conjecture, which has
been recently proven, provides this threshold for all alphabet
sizes. Motivated by a question of Krieger, we deal here with
the analogous threshold when the infinite word is restricted to be a D0L
word. Our main result is that, asymptotically, this threshold
does not exceed the unrestricted threshold by more than a little.