Not sticking to 1 months' notice period

Other

My tenant (who claims housing benefit) has been offered a council house.
I have written to her to state that she must give 1 months' notice in writing. This would be effective up until 9th August. The problem is, she doesn't know what date the new property will become available.
If it is before the end of the months' notice, the council may not pay benefits to cover both properties. If it is after the end of the month how am I covered legally, as the contract will be deemed to have finished?