We consider the one-way ANOVA problem of testing the equality of several normal means when the variances are not assumed to be equal. This is a generalization of the Behrens-Fisher problem, but even in this special case there is no exact test and the actual size of any test depends on the values of the nuisance parameters. Therefore, controlling the actual size of the test is of main concern. In this article, we first consider a test using the concept of generalized p-value. Extensive simulation studies show that the actual size of this test does not exceed the nominal level, for practically all values of the nuisance parameters, but the test is not too conservative either, in the sense that the actual size of the test can be very close to the nominal level for some values of the nuisance parameters. We then use this test to propose a simple F-test, which has similar properties but avoids the computations associated with generalized p-values. Because of its simplicity, both conceptually as well as computationally, this F-test may be more useful in practice, since one-way ANOVA is widely used by practitioners who may not be familiar with the generalized p-value and its computational aspects.