If the debt occurred from a joint tax return that you both filed, you probably can't. Regardless of whose income caused the balance due, both persons are equally responsible for it if a joint tax return is filed.
There is a provision in the tax code for tax relief as an "Innocent Spouse".
It can be quite difficult to prove but it may be an option.
See a local tax professional.
There is also something called an "Offer In Compromise" where you settle the debt for less than you owe if you can not possibly ever pay it off.
Again, see a local tax professional.
Wayne Z said it right. If you signed a joint return with any of the years taxes are owed, you are responsible also for payment. If he filed them, for example, before you married him, they can drop off if he has no assetts. His estate on the final return will apply money to the debt starting with the oldest debt first.
Death does not discharge debt.
Pay it off.