Hmm - can you impose a date for proof of mortgage/intent?
Personally, I'd by inclined to accept their offer but tell them that you will not be taking the house off the market until they have fully committed - and that if you get another/better offer you will accept it. If they really want it, then this may be enough to spur them into action. (Is this legal?)
You could accept the offer but with the proviso that your house stays on the market until they are in a position to proceed. We were messed around in the same way by the people we were buying from and in the end pulled out so know where you are coming from!
sounds to me like they pulled out perhaps getting a hint that the market was taking a downturn. did they make a lower offer that you refused the first time?
in any case I agree with others - accept but with no guarantees and tell the agent not to put 'under offer' on the details or tell other prospective viewers, it puts people off. you could get all huffy and say no but that leaves you in a position you don't want to be in.
I'm normally very anti-agzumping but I think in a situation like this where someone has already dropped out once then its fine provided you tell them that you will accept another offer if they can exchange quicker.
I would hold out longer than mortgage offer though. I would keep marketing until solicitors say they are on the verge of exchanging.
Oliveoil - Make sure both parties know about the other party waiting in the wings. It will make them less likely to muck about and try for any further reductions after the survey etc. It's worth a try to get the new buyers to match the offer that is on the table, isn't it?