Homelite string trimmer will not run

I got an old Homelite string trimmer (UT20046) from my neighbor this year. I've replaced every part that I know how to replace. Fuel filter, fuel lines, air filter, spark plug, even the entire carburetor. It will start and run for about a minute. After that, it won't even start. Any ideas on what could be wrong? Should I just eat it and get a new one?