We present and analyze a model for the interaction of human immunodeficiency virus type 1 (HIV-1) with target cells that includes a time delay between initial infection and the formation of productively infected cells. Assuming that the variation among cells with respect to this 'intracellular' delay can be approximated by a gamma distribution, a high flexible distribution that can mimic a variety of biologically plausible delays, we provide analytical solutions for the expected decline in plasma virus concentration after the initiation of antiretroviral therapy with one or more protease inhibitors. We then use the model to investigate whether the parameters that characterize viral dynamics can be identified from biological data. Using non-linear least-squares regression to fit the model to simulated data in which the delays conform to a gamma distribution, we show that good estimates for free viral clearance rates, infected cell death rates, and parameters characterizing the gamma distribution can be obtained. For simulated data sets in which the delays were generated using other biologically plausible distributions, reasonably good estimates for viral clearance rates, infected cell death rates, and mean delay times can be obtained using the gamma-delay model. For simulated data sets that include added simulated noise, viral clearance rate estimates are not as reliable. If the mean intracellular delay is known, however, we show that reasonable estimates for the viral clearance rate can be obtained by taking the harmonic mean of viral clearance rate estimates from a group of patients. These results demonstrate that it is possible to incorporate distributed intracellular delays into existing models for HIV dynamics and to use these refined models to estimate the half-life of free virus from data on the decline in HIV-1 RNA following treatment.