Why is it reasonable to believe that this student handed in a term paper from another class?
I might be having a completely different stance on the topic, however I feel it needs to be thought about. The primary goal of a school would be to asses whether the student in question has achieved a certain level of competences. 
In order to test those competences you'd give all sorts of assignments to see whether the student has both competences AND skills required for their studies.

As @Significance and @DanRomik mentioned before, filing the other paper can be seen as self-plagiarism. However, if you consider that filing this paper is invalid you could also say that he never filed any valid paper in the first place. Meaning he crossed the deadline that was given for this paper.

School is a place to prepare a person to work at companies and more. So it would not be a bad idea to treat a student as if he were an employee. He filed his work late, couldn't meet the deadline and would obviously be penalized. Now since school is a place to learn from your mistakes you could throw him a lifeline. Rather than smashing him down, I'd accept his paper that was of extremely poor quality, but penalize the grade of this paper based on crossing the deadline.

Now, in the end he may still fail due to the poor quality + crossed deadline. But at the very least he would have learned to be more punctual about his work. 

After all, being able to deliver good quality work on a timely way are some of the most important competences a student has to learn.