If you eneable Backface Culling from the Properties panel you will see that half or your heart has its face normals pointing inwards.
Also the edges you are trying to bevel have duplicted Vertices.
To fix your beveling you need to do as M.ardito suggested:
With all of the heart selected Recalculate Normals and Remove Doubles
If you now select that edge and use the Bevel tool it will look alot better, but.... if you look closely there is still a little problem area.
To correct that anomalie Patdog has given the solution,
Select only the top faces of the heart and using the Insert tool (keyboard shortcut key I ), insert a inner edge loop on those faces.
Now you can go back to using the Bevel tool and it should bevel ok.
The extra edge loop on the top face can now be deleted ( X > Edge Loops )