Context. A steady model was presented by Burn, in which energy conservation is used to constrain the parameters of stochastic Fermi acceleration. A steady model, however, is unlikely to be adequate for particle acceleration in impulsive solar flares. Aims. This paper describes a time-dependent model for particle acceleration in a Fermi reservoir Methods. The calculation is based on the original formulation of stochastic acceleration by Fermi, with additional physically motivated assumptions about the turbulent and particle energy densities within the reservoir, that are similar to those of the steady analysis. The problem is reduced to an integro-differential equation that possesses an analytical solution. Results. The model predicts the formation of a power-law differential energy spectrum N(E) similar to E-2, that is observable outside the reservoir. The predicted spectral index is independent of the parameters of the model. The results may help in understanding particle acceleration in solar flares and other astrophysical applications.