Must a single-borrower mortgage be totally paid off on death, or could it be refinanced for the widow?

My husband's name is the only one on the Mortgage/Note, but both of our names are on Title and the house is in our Trust. He is giving his half to me and not his children upon his death. What happens with the Note on his death? I am paying the mortgage now from my account. The Trust divides his other assets among all of us, and my portion COULD pay if off, if necessary, but is it?.