This is a dilemma for most project managers in small to medium sized companies. Since such companies cannot afford to have full time project managers, project managers are expected to actually accomplish some tasks and to consider themselves as resources in the project.
My opinion is to first have a little chat with him about the situation. Make sure you don't point fingers and you don't sound aggressive, just explain the situation in very objective terms and ask him what his solution to the problem might be. Do not suggest the solution yourself before asking him about his solution.
It is normal for the person to become defensive (I have so much work, this company sucks, etc...) when this talk happens - but it is very important for you to keep your cool. Remember, this is a person and not a machine.
His solution might not be reasonable or feasible or practical, so when this happens, you can suggest your solution and ask him about his opinion (for example, throwing another resource on his task, delegating his whole task, etc...) If he maintains a negative attitude, then you should go ahead with your own solution anyway. You should also send an email to the project sponsor (and email that doesn't sound naggy) that you have assigned or delegated the task to someone else since that person is overloaded.
Always remember to maintain a professional attitude all through the process - and never, ever complain to your project sponsor! (let him complain and then you can explain your point of view)