Try attaching it to his collar when in the house and just let him drag it around to get used to it. Make sure the leash is lightweight and doesn't make a lot of noise -- he needs a regular puppy leash right now, not a heavy adult leash or a retractable leash.
Also don't give in and pick him up when you're outside with him -- it only reinforces the behavior because he gets what he wants. Ignore him for the most part and try to coax him to walk with you while the leash is kept loose. Don't "comfort" him or baby him since that will also reinforce his fear -- just act happy as if there is nothing to be concerned about. He'll come around eventually
