You do have a lease, and it is a month to month oral lease. It appears in NC that you only need to give 7 days written notice to terminate this type of tenancy. See: http://www.nolo.com/legal-encyclopedia/north-carolina-notice-requirements-terminate-month-month-tenancy.html . And: § 42-14. Notice to quit in certain tenancies.
NC General Statutes - Chapter 42-4
A tenancy from year to year may be terminated by a notice to quit given one month or more before the end of the current year of the tenancy; a tenancy from month to month by a like notice of seven days;
Read more: http://www.city-data.com/forum/renting/827506-nc-law-burden-notice-end-tenancy.html#ixzz2YMshuZTL BE SURE AND GIVE HER WRITTEN NOTICE TO VACATE! If she doesn't vacate at the end of 7 days, then you will need to file an eviction suit in your local eviction court. They should have instructions online or at the filing desk - or contact a local attorney to help you.
To terminate a month to month lease, you will need to give the tenant 7 days notice in order to make that coming month be the last month on the lease. You won't get the tenant out in 7 days. In fact, you'll have to sue in small claims court to the get the tenant out and you won't be able to do that until the tenant breaches. Based on the question, it would appear that your earliest opportunity to demonstrate a breach would be for the tenant to remain in possession after the end of the next month...assuming the tenant is paying as she has been.
It is problematic for you that you are posting your question stating you gave a one year lease. To your benefit most magistrates in small claims court aren't particularly pro-tenant to start with, but from a legal point of view, if you did in fact make a year long lease, then you have big problems here. So, I would recommend you sort that out and start being very careful in your statements about the tenancy.
I would also strongly recommend that you hire an attorney to assist you with this case because by virtue of your tenant being in the process of buying a HUD home, it would appear that she will be able to qualify for free legal assistance from Legal Aid of North Carolina. An "average" Legal Aid attorney would rip you to shreds on this case with no lease, no clarity about the term and the fact that you think you are going to move in new tenants on August 1 when you haven't even initiated any legal proceedings to get the current tenant out yet.
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I agree with the previous answers and in particular Mr. Tate's analysis of the realities of NC law. To that effect, it appears you are trying to maximize how much you can make by having a tenant in immediately after another tenant has left.
If she leaves July 1st, great you have not problems. If she doesn't then you will. If you choose to keep her then, you run the risk of her stringing you along. Further, I am not sure what she means by buying a HUD home. HUD does not sell home, they issue subsidies to rent homes or apartment. Perhaps she was referring to Habitat for Humanity or another similar service agency?
In any event, since she is paid up through July, you need to send her notice at least 7 days before the end of July to be out on the 31st. If you want her out, then you probably should send the notice ASAP. From there, if she doesn't leave you have to go to small claims court and evict her. Assuming you have the hearing, you win, she does not appeal, and there are no delays, you are looking at having the home ready to move no sooner than approximately the 25th of August assuming you don't have to do too much to prepare it for the next tenant.