1) The assumption of equality of variance is almost never exactly satisfied
2) ANOVA isn't very sensitive to mild differences in variance, so your ability to reject the null in the Levene test may be almost irrelevant in large samples; correspondingly an inability to reject the null in small samples should be no comfort whatever. Hypothesis tests are simply answering the wrong question [The question: "do the variances differ?" ... to which the correct answer is almost always 'yes, they do' ... is almost useless, since we know the answer already, and a hypothesis test only tells us what we already know in sufficiently large samples. A much better question is How much does that affect my inference? -- which is not answered by a hypothesis test; it's more related to the relative sizes of the variances, and to the relative sizes of the samples they occur in].
3) If sample sizes are equal (or very nearly so), then the ANOVA isn't sensitive to the assumption.
A better choice than testing variance and then choosing what analysis to apply based on the outcome of that test is to not assume equal variances at all (unless you have a reason to do so). A Welch-Satterthwaite type approach works pretty well - just use it whenever you don't have a good reason to do otherwise.