Fourteenth Amendment, Section 4:
The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned. But neither the United States nor any State shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.

The debt limit is unconstitutional. Congress already authorized the debt when they passed last years' budget. The 14th amendment declares that the validity of that debt shall not be questioned, and any legislation preventing payment of that debt is unconstitutional. 
The Treasury department should just declare that the 14th amendment renders the debt ceiling unconstitutional and they will continue to make interest payments on the federal government's debts.