It is well known that maximum connectivity graphs play an important role in the design of reliable networks. The class of symmetric graphs called circulants is known to contain such maximum connectivity graphs. Although not all circulants have this maximum connectivity property, those that do have a great variation in their diameters. Since diameter is a measure of transmission delay, the minimum diameter, maximum connectivity circulant problem is considered here. Several results are given for the general case and a simple solution is derived for the connectivity four case.