We examine, in detail, the conditions under which the quark-hadron phase transition in the early Universe can influence the previously determined upper bound on the axion decay constant f/sub a/< or approx. =5 x 10/sup 12/ GeV. By integrating the axion equation of motion (in general and in a specific model) through the phase transition we find that if the phase transition is sufficiently abrupt the limit on f/sub a/ may change by an order of magnitude in either direction (depending on initial conditions); the most probable result is a tightening of the bound by a factor approx.5. Sensitive axion detectors, as are now being contemplated, could conceivably provide information on the Universe at approx.10/sup -5/ sec after the big bang.