Purchased new contract on old account. Best way to transfer internet service with little downtime?

I just purchased a new contract (with new account number) under my old profile, which already has internet service on a seperate account.  My profile now has 2 accounts under 1 address.  The plan is to cancel the current old account (preferably before the next billing cycle kicks in) and switch service to the new account.  I was wondering the best way to transfer internet service with as little downtime as possible?

My wife works remotely, so I was hoping to have it done in an afternoon/evening or weekend, as soon as possible.

I am using my own modem.

Any info is greatly appreciated.