This paper studies the rank of the divisor class group of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, there is a small modification of Y with a second extremal ray; Cutkosky, following Mori, gave an explicit geometric description of contractions of extremal rays on terminal Gorenstein 3-folds. I introduce the category of weak-star Fanos, which allows one to run the Minimal Model Program (MMP) in the category of Gorenstein weak Fano 3-folds. If Y does not contain a plane, the rank of its divisor class group can be bounded by running an MMP on a weak-star Fano small modification of Y . These methods yield more precise bounds on the rank of Cl Y depending on the Weil divisors lying on Y . I then study in detail quartic 3-folds that contain a plane and give a general bound on the rank of the divisor class group of quartic 3-folds. Finally, I indicate how to bound the rank of the divisor class group of higher genus terminal Gorenstein Fano 3-folds with Picard rank 1 that contain a plane.