Yep, and it is usual.
Believe it or not you cannot just get rid of the cleaner, which is ridiculous considering that is probably the reason why the customer wants to quit the old contract. If they have been there under a year your task is easy though, if not you will have to be very careful!
You will need proof from the cleaner of what her hours and pay is, you are also responsible for any holiday due!!!
When do you take over the contract?