Data from settings in which an initiating event and a subsequent event occur in sequence are called doubly censored current status data if the time of neither event is observed directly, but instead, it is determined at a random monitoring time whether either the initial or subsequent event has yet occurred. This paper is concerned with using doubly censored current status data to estimate the regression coefficient in an accelerated failure time model for the length of time between the initiating event and the subsequent event. Motivated by a problem in AIDS epidemiology, attention here is focused on a special case, the case in which the initiating event, given that it has occurred prior to the monitoring time, may be assumed to follow a uniform distribution. The main result is that the likelihood in the special case has the same structure as the likelihood in a simpler setting, the setting in which the time of the initiating event is known. The result allows methods developed for the simpler setting to be applied in the special case. The results of application of the approach to real data is reported.