:( I found out the mower is a Lawn Chief. But it will not run for very long. The carburator has new parts, the transmission is fine, but when it is started it will quit as soon as it is put in the run or idle position. Think it may be the motor, but any ideas on what part?

Very well the carb is not set right, 2 adjusting screws, one for idle, one for high speed, both turned all the way in, then backed out 1.5 turns to get it going enough to tune further, also the needles themselves may be worn out, but you never gave engine brand, nor horse power rating so it could very well be a diapragm carb. And the diapragm is worn out or folded. Its not the engine! I see many thrown away because people thought it was the engine, when it very well turned out to be a cheap part, or a misadjustment or dirty carb, hense also, was the carb cleaned out good, and I mean every tiny hole, if it's a float carb, most of the time the inlet needle gets stuck or is worn out, or the main jet right under where the float bowl nut goes into is clogged.

Check the governor. The carburetor screws should still be where they should be unless an unskilled person turned them. The governor actually adjusts the butterfly in the carburetor. Often they were just a metal plate in the air stream of the fins in the flywheel. It may have frozen up over time as it sat around. The plate needs to spring back and open the throttle as the speed drops under a greater load. Without more gas, the engine will die.

My father changed the diaphram and gaskets. It seems that the carb might be warped and the vacuum is broken. How can I find a new carb and tank for this thing?? There are no markings other than model and serial #s. I really like it and don't really want to have to trash it. :o

Forget Lawn Chief. The Briggs and Stratton engines are very common. Most small engine parts suppliers should be able to work from the engine model and serial number. The first one you talk to, may not be interested. Try the small, independent shops. You might try to find B&S parts on the net.

As for the carb, you can upgrade to new, and find a used newer tank