Unfortunately, you will actually have to sell him the house. The important thing to remember is that the mortgage company has a lien on the property, and that lien will have to be satisfied before you are allowed to transfer the property. Otherwise, people could easily circumvent a mortgage and a mortgage company would have no recourse.
Depending on how much equity you have in the house, here is one possible way of resolving your situation. Assume that you have a house worth $100K with a $80k mortgage left over, meaning you have $20K of equity in your property. You could likely get a line of credit against your home equity, borrow $20k, and then transfer the $20k to your son. Then, your son could obtain a $100k mortgage on the property and put the $20k down, and you could sell the property to him for $100k, which would be enough to satisfy your mortgage and satisfy the $20k loan
you took out against your property. Another option is for your son to approach a bank and inform them of the situation and how the property already has equity built into it.
Overall, you may have options, but the mortgage company does not have to agree to work with you and you would be responsible for satisfying the mortgage before you could transfer the property's title