So it seems this is a custom for tenants... you're not alone. People these days have trouble honoring their word (in this case, their contracts). So, be nice... but don't be too nice to put yourself in the possition for the entire world to screw you over.
If I were you... Send him a written notice (via e-mail,if available, and certified mail) of how much he owes you for this month's rent, state how much it would be for late fees, and remind him when the full rent is due. This usually makes them realice that you are serious about getting your full month's rent, PLUS its an additional proof that no deal was reached with him to pay a pro-rated rent.
When the last day of grace period according to your contract comes up (mine says rent is fue on the 1st with a grace period of 5 days), if he still hasn't paid, get ready to give him a three (3) day notice to pay or leave... that usually makes them pay.
If that did not make him pay file an eviction.
I wonder if tenants do the same with their car or credit card payments... I bet they try their best not to, because they know they'll be screwed. Then I don't understand why it should be any different with the house payments. I say: Pay or leave brother... This is fair. Good luck!
HOWEVER, if you're this really nice person, and don't really need the money for rent (at least not very urgent)... you may consider helping him out. DO ask him for a copy of the new lease where he's moving to (just so you can rest asure that he WILL be leaving when he's telling you) or a statement saying he will be leaving when he says he'll be leaving... and pray everything goes well... if you do this, then this is what we call mercy. Good luck either way!