2001 Saturn SL2 - engine problems?

Our 2001 Saturn SL2 stopped without a sputter yesterday, then was fine after a jump start. I made it about 20 miles to work; when I left, it stopped again at a light. We thought the battery died so we got a new battery - the lights in the car came on but the engine wasn't turning over.

A mechanic told me it could be the engine, but they don't seem sure. Yeah, I know get a new mechanic, right. My oil was low but we change it religously. The mechanic said he thought it possibly could be the starter.

He wants to charge us $3,800 for the engine and labor, although he said he "thinks" that's what it is.

The check engine light has never come on, and the there were no snaps, bangs, etc. From the engine.
Some smoke did come out from under the hood when we tried to start it with the new battery.

Now we're wondering if it isn't the computer?
Any opinions?
Help!