Also bear in mind that
One-factor anova is "a generalization of the
two-sample t-test." The 2-sample t-test can deal with unequal variances, which leads me to think that so can ANOVA.
 Quote by Engineering Stat. Handbook, sec. 1.3.5.3 (Two-Sample t-Test for Equal Means)
The variances of the two samples may be assumed to be equal or unequal. Equal variances yields somewhat simpler formulas, although with computers this is no longer a significant issue.
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But, a particular software package may not have this option. In some ANOVA packages (like SAS) specifying the "Welch" option controls for unequal variances.