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We report evidence that the asymptotic low-energy power-law slope a (below the spectral break) of BATSE gamma-ray burst (GRB) photon spectra evolves with time rather than remaining constant. We find that a high degree of positive correlation exists between the time-resolved spectral break energy Epk and a. In samples of 18 “hard-to-soft ” and 12 “tracking ” pulses, evolution of a was found to correlate with that of the spectral break energy Epk at the 99.7 % and 98 % confidence levels, respectively. We also find that in the flux rise phase of hard-to-soft pulses, the mean value of a is often positive, and in some bursts the maximum value of a is consistent with a value. 11. BATSE burst 3B 910927, for example, has an amax equal to 1.6 H 0.3. These findings challenge GRB spectral models in which a must be negative or remain constant