What to do when data fail tests for homogeneity of variance (part of one-way ANOVA)?

One-way ANOVA assumes that the data come from populations that are Gaussian and have equal variances. GraphPad Prism tests this assumption with Bartlett's test.

Similarly, the unpaired t test assumes that the data are sampled from Gaussian populations with equal variances, and GraphPad Prism tests this assumtpion with an F test.

If these tests result in a small P value, you have evidence that the variance (and thus standard deviations) of the groups differ significantly.

It is not really clear what to do next. Here are some thoughts:

• This gives you strong evidence that the groups are not selected from identical populations. You haven't yet tested whether the means are distinct, but you already know that the variances are different. This may be a good stopping point. You have strong evidence that the populations the data are sampled from are not identical.
• Some statisticians suggest never using Bartlett's test. It is too sensitive to minor differences that wouldn't really affect the overall variance. So if the difference in variances is not huge, and especially if your sample sizes are equal (or nearly so), you might be safe just ignoring Barlett's test.
• Some suggest using Levene's median test instead. Prism doesn't do this test (yet), but it isn't hard to do by Excel (combined with Prism). To do Levene's test, first create a new table where each value is defined as the absolute value of the difference between the actual value and median of its group. Then run a one-way ANOVA on this new table. The idea is that by subtracting each value from its group median, you've gotten rid of difference between group averages. (Why not subtract means rather than medians? In fact, that was Levene's idea, but others have shown the median works better.) So if this ANOVA comes up with a small P value  then it must be confused by different scatter (SD) in different groups. If the Levene P value is small then don't believe the results of the overall one-way ANOVA. See an example on pages 325-327 of Glantz.
• Don't be too quick to switch to using the nonparametric Kruskal-Wallis ANOVA (or the Mann-Whitney test when comparing two groups). While nonparametric tests do not assume Gaussian distributions, the Kruskal-Wallis and Mann-Whitney tests do assume that the shape of the data distribution is the same in each group. So if your groups have very different standard deviations and so are not appropriate for one-way ANOVA, they also should not be analyzed by the Kruskal-Wallis or Mann-Whitney test.
• Often the best approach is to transform the data. Often transforming to logarithms or reciprocals does the trick, restoring equal variance.
• Read more about the general topic of assumption checking after ANOVA in this article by Rich Ulrich,  this section of the Prophet StatGuide, or here.