(1) "The means of each group are normally distributed" - on what basis can you make such an assertion?

(2) your difference in variance sounds pretty small, and if sample sizes are nearly equal would cause little concern, as others have mentioned,

(3) Welch-type adjustments* for degrees of freedom exist for ANOVA just as with two-sample t-tests; there's little reason not to use them as a matter of course. Indeed, the oneway.test function in R does this by default.

*B. L. Welch (1951), On the comparison of several mean values: an alternative approach.
Biometrika, 38, 330–336.

(1) "The means of each group are normally distributed" - on what basis can you make such an assertion?

(2) your difference in variance sounds pretty small, and if sample sizes are nearly equal would cause little concern, as others have mentioned,

(3) Welch-type adjustments* for degrees of freedom exist for ANOVA just as with two-sample t-tests; and just as with their use in two sample t-tests, there's little reason not to use them as a matter of course. Indeed, the oneway.test function in R does this by default.

*B. L. Welch (1951), On the comparison of several mean values: an alternative approach.
Biometrika, 38, 330–336.