Differential phase contrast computed tomography (DPC-CT) is an x-ray imaging method that uses the wave properties of imaging photons as the contrast mechanism. It has been demonstrated that DPC images can be obtained using a conventional x-ray tube and a Talbot-Lau-type interferometer. Due to the limited size of the gratings, current data acquisition systems only offer a limited field of view, and thus are prone to data truncation. As a result, the reconstructed DPC-CT image may suffer from image artifacts and increased inaccuracy in the reconstructed image values. In this paper, we demonstrate that a small region of interest (ROI) within a large object can be accurately and stably reconstructed using fully truncated projection datasets provided that a priori information on electron density is known for a small region inside the ROI. The method reconstructs an image iteratively to satisfy a group of physical conditions by using a projection onto convex set (POCS) approach. In this work, this POCS algorithm is validated using both numerical simulations and physical phantom experimental data. In both cases, the root mean square error is reduced by an order of magnitude with respect to the truncated analytic reconstructions. Truncation artifacts observed in the latter reconstructions are eliminated using the POCS algorithm.