Well, if they are planning to stay then this cannot be their "last" month. The original contract ended by its own terms after that first year, so yes, it is over. Their unilateral decision that July 2013 is their last month and thus they don't have to pay rent would be fine - IF they were planning on moving out before the end of the month.
You don't need to sign a new lease with them if you are OK with the month-to-month arrangement. If you don't have a written rental agreement, you cannot take a security deposit (see RCW 59.18.260) and no, this isn't a "new" agreement such that you'd take a new deposit and a new last month's rent.
In order to be sure you are managing things properly, take your contract to a local real estate attorney who understands residential tenancy law in WA, and get the advice you need.
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I agree with Ms. Powell and the May would the penultimate month and if they are staying then they still owe for June. if they were leaving, then the story would be different.
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I agree with my colleagues that this is likely the best interpretation - unless you have some terms in your lease I have never seen.
If they do not move out and accept rent then the tenancy continues month-to-month without any further written agreement. On these facts, if you do not want to allow the last months rent deposit to be applied you should inform them in writing that this in not acceptable ASAP. You need to insist they pay June before you accept July rent (or any other rent) - unless you are willing to live with not have a deposit going forward.
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