Smells, it depends on what kind of loan you have. A regular FHA loan, FNMA or FreddieMac fixed rate or ARM mortgage allows for one of the borrowers to assume the loan at it's current terms if the remaining borrower credit qualifies and income qualifies. If it is an FNMA or Freddie loan and has private mortgage insurance, the MI company may have to approve the assumption of the loan by the remaining borrower.
Call the phone number for the Servicing department where you make you make your payments and tell them you want to assume the loan and release the co-borrower from liability. Most of the time the problem arises because the no one in the Servicing department knows how to process the loan assumption. There will be a cost for this.In my previous life, it was about $500 for an FHA and 1% of the unpaid balance for the FNMA or FreddieMac loans. That may not be the same for all Servicers so you'll need to ask.
VA loans also allow this, but if the borrower that is being released is the veteran that used their entitlement, they don't get that portion of their entitlement restored.