We present X-ray light curves (1.5-12 keV) for fifteen gamma-ray bursts (GRBs) detected by the All-Sky Monitor on the Rossi X-ray Timing Explorer. We compare these soft X-ray light curves with count rate histories obtained by the high-energy (>12 keV) experiments BATSE, Konus-Wind, the BeppoSAX Gamma-Ray Burst Monitor, and the burst monitor on Ulysses. We discuss these light curves within the context of a simple relativistic fireball and synchrotron shock paradigm, and we address the possibility of having observed the transition between a GRB and its afterglow. The light curves show diverse morphologies, with striking differences between energy bands. In several bursts, intervals of significant emission are evident in the ASM energy range with little or no corresponding emission apparent in the high-energy light curves. For example, the final peak of GRB 970815 as recorded by the ASM is only detected in the softest BATSE energy bands. We also study the duration of bursts as a function of energy. Simple, singly-peaked bursts seem consistent with the E^{-0.5} power law expected from an origin in synchrotron radiation, but durations of bursts that exhibit complex temporal structure are not consistent with this prediction. Bursts such as GRB 970828 that show many short spikes of emission at high energies last significantly longer at low energies than the synchrotron cooling law would predict.
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