How do you fade the edges of a model?

I have a model that is made of a flat plane that has been sculpted in the middle. I need to fade the edges of the plane gradually into transparency. How do I accomplish this? I am totally new to this! Thank you for help!

Edit:

Response to Cegaton:

Thank you, this works just as described on a new plane. It is really just what i need, but it does not work on my model, which was modeled in Blender 2.7, if that makes any difference. On my model, the adjustment only affects the top half of the mesh, and the bottom part is either translucent or opaque, with sharp edges. I am baffled. I will add that the model was made from a plane but many nips and tucks and extra rows of vertices have been added since the beginning. I wonder if there is something about an older model that does not work with the new compositing options ('separate xyz' was not available in 2.7).

PROBLEM SOLVED: The transparency begins at model's origin. I moved the origin point from the center to the bottom of the model and now the transparency adjustment is operating exactly as demonstrated in Cegaton's answer.

Suppose you sculpted this beauty out of a plane:

To make the flat area transparent you can create a node setup like this, where the transparency is controlled by the height (Z axis) of the model.

EDIT:

If you are working on an older version of blender that doesn't have the separate XYZ you can use a separate RGB node:

For finer control on the transparency use a Vector->Mapping node. By moving the Location values on the Z axis you can determine the height where the transition starts, and by moving the sliders on the color ramp you can determine how smooth the transition will be.

• Thank you, and I have a question. I posted it as an edit to the original question because it was too long to be a comment. Sep 7 '15 at 6:16
A cycles material where the Diffuse node could be replaced with your own shader tree. This method uses the distance x,y from 0,0,0 (ignores the z component) - but does need the Separate XYZ node.