This paper presents a relatively simple example of a bifurcation of a central configuration in the four body problem. There have been several indications in the literature that bifurcations can occur, but no one has carried the analysis to completion. Our goal is to give a simple proof that the bifurcation that they observe actually occurs in the full 4 body problem with one small mass. First we give a mathematical proof of a theorem that gives conditions when a degenerate critical point of the restricted problem can be continued into the full 4 body problem as a degenerate central configuration. The theorem proves that the degeneracy is actually due to a bifurcation provided other partial derivatives of the potential be non-zero. This is a special case of a theorem in catastrophe theory. Finally we use numerical methods to verify the hypothesis of the theorem in the restricted 4-body problem.