Just because you give two weeks notice, that does not mean that they have to keep you as an employee for those two weeks. They can fire you or agree to accept your resignation immediately to save money. To be on the safe side and not tick off your employer for letters of reference in the future, and to be sure that you have July covered in insurance, I would give your two weeks notice on July 1st and either work those 2 weeks or hope that they just tell you that you are free to leave.
Mr Leroi's answer is spot on. In addition, although most company's pay the two weeks, there is no law requiring them to do so. Many employers who think they are being used will walk the employee to the door the monument they quit and not pay the two weeks. It is always bad form to quit without notice, but there is no law requiring notice either. You actions should be governed by your relationship and your need for future references. If you have a good relationship take it straight up, give the notice and hope the company will keep you there to July 5th. In fact, be prepare to argue why they should keep you there. Good luck.
Answering this quetion does not establish an attorney client relation. The answer is for educational purposes only. You should consult an attorney for your circumstances.