A derivative-free Quasi-Newton (DFQN) method previously published [J. Greenstadt, Math. Comp. , v. 26, 1972, pp. 145-166] has been revised and simplified. The main modification has the effect of keeping all the successive approximants to the Hessian matrix positive-definite. This, coupled with some improvements in the line search, has enhanced the performance of the method considerably. The results of numerical trials on many of the "standard" test functions are displayed, in addition to comparisons with two other methods. These indicate that the present DFQN method is not too far behind that of Gill, Murray and Pitfield, the most efficient one presently known.