Creating a smooth spherical surface I am designing some parts for 3D printing and I have been banging my head for several days already trying to accomplish the following:

I have a "frame" that looks like this:


I am trying to create a surface inside the frame that has a nice organic, smooth spherical shape similar to the top shell of a computer mouse. 

Here is a close illustration of what I am trying to get:


Unfortunately I am unable to achieve an acceptable result. I have tried curving a surface, using NURBS surface with modifiers, but still nothing.

What are the options for achieving this and having optimal control over the curved surface profile while it still clamps exactly to the "frame"?

UPDATE:
Here is my blender file with the frame that needs to have the surface along with a sample curve that I want for a profile for the surface.



Any suggestions?
Try extruding the shape, then use the bevel modifier with limit method to weight. By fine tuning the edge bevel weight in edit mode on the top edges you should be able to get something close to what you want.