It is important to note that what the debt ceiling actually is, is the bill for spending that has already taken place coming due for payment, not the initiation of new spending. So there is no conflict between which law Treasury should follow, because that decision was already made some time in the past based on the Full Faith and Credit of the United States. The money needed to implement all spending, not some of it, has already been spent (on credit) and the debt limit being reached is just a marker of when that money physically leaves the Treasury's account. The debt limit is the statutory limit of borrowing that the Treasury is allowed.
In fact, the United States would have reached the limit sometime in December, however, the Secretary of the Treasury Tim Geithner has been making prioritized decisions for awhile now in order to extend the deadline as far as possible. That is, he has stopped funding discretionary things within the federal government that don't go to outside creditors so that there is more money on hand to pay those creditors as their bills come due.
There are logistical problems with prioritizing payments further as Senator Toomey would like to mandate that @DVK did a good job of laying out in his answer. However, most observers think that the government would still default without a raise in the limit, because the payments owed in March (when the limit is expected to be reached) do not all come due at the end of the month. Most are due on the 15th, and the income tax revenues the Treasury takes in each day very wildly. While the United States takes in enough in a month to pay these minimum payments, it is unlikely that we would do so early enough in the month to make all payment obligations.