Keep in mind that an employer also looks at how you are leaving your previous position. If it looks like you are, on any level (intentionally or not) leaving your current employer in the lurch or violating an employment contract, it won't look good for you.
If you really want to apply, make it clear that you are NOT available until July 1st. Be up front, and make it clear that you are dedicated to your current employer and the terms of your contract/grant. If you're honest and are worth their time, they'd be willing to push the start time forward.
Live as if you were to die tomorrow. Learn as if you were to live forever. - Mohandas Gandhi