Conditional acceptance
Every such case is individual and depends on the rules set in place. Contact both universities, actually. Perhaps your current institution can assure the new one that the diploma is certain to be issued by some date. Perhaps the new institution will accept that. But it is up to them. 

I doubt that they will be especially finicky about the rules, but no one can say but them. Keep all of your other options open until things are finalized. 

Even a letter from a professor might assure them that all is well.