The so-called Jacobi Procedure is extended to the case of normal matrices. A stable iterative procedure is described utilizing plane unitary transformations for such matrices which yield both the characteristic values and their associated vectors. Generally, the technique consists of minimizing at each stage the sum of the squares of the off-diagonal elements of the given matrix; however, there is one case in which this leads to no improvement; i.e. the lowest value for the change is non-negative. In this case, it is shown that a convergent procedure is still possible.