Let $f:X\to Y$ be a generically finite proper morphism of varieties. There is some locus in $Y$ over which the fiber of $f$ is positive dimensional, so we blow it up, along with the preimage of it in $X$ to get a map $\tilde{f}:\tilde{X}\to\tilde{Y}$ which has finite fibers.

Are there any nice conditions that will guarantee that the map $\tilde{f}$ is flat?

genericallyfinite? I think any finite morphism is already quasi-finite, i.e., has finite fibers. – Dave Anderson Aug 3 '11 at 18:52