  We calculate the rate of in-fall of stellar matter on an accretion disk
during the collapse of a rapidly rotating massive star, and estimate the
luminosity of the relativistic jet that results from accretion on to the
central black hole. We find that the jet luminosity remains high for about 100
seconds, at a level comparable to the typical luminosity observed in gamma-ray
bursts (GRBs). The luminosity then decreases rapidly with time for about 10^3
seconds, roughly as ~ t^-3; the duration depends on the size and rotation speed
of the stellar core. The rapid decrease of the jet power explains the steeply
declining X-ray flux observed at the end of most long duration GRBs. A X-ray
plateau is also produced by continued fall-back of matter -- either from an
extended stellar envelope or from material that failed to escape with the
supernova ejecta. In a few GRBs, the X-ray light curve is observed to drop
suddenly at the end of the plateau phase, while in others the decline is ~ t^-1
- t ^-2. These features arise naturally in the accretion model depending on the
radius and mean specific angular momentum of the stellar envelope. The
accretion model thus provides a coherent explanation for the diverse and
puzzling features observed in the early X-ray light curves of GRBs. (Abridged)
