It will fit and bolt right up. I sold a 63 Coupe with a 62 engine in it.
While it would physically fit, keep in mind the 63 390 was a completely redesigned engine. The 1962 and prior 390 is a slightly taller and longer engine block so may seem a bit more cramped, and distributor is at the back making maintenance more cumbersome. If at all possible I would try to find a 1963 390 (or 1964 Hydramatic 429) to go back in it, but understand you have to use what is available.
Here is a summary of some of the changes as noted in our Engine and Transmission Facts - differences from 62 to 63 to 64 to 65 help topic:
The 1963 390 was a totally redesigned engine compared to the 62 and earlier 390's. It was 50+ lbs lighter, engine accessories including the distributor was moved to the front and used a new aluminum engine cover that incorporated the oil pump. Nearly 1/2" was shaved off the top of the block, connecting rods were shortened to keep the same compression ratio, block length was shortened by more than an inch. They switched to a hollow-cast crank, which is 11 pounds lighter than the 1962 crankshaft, and switched to an alternator-rectifier charging system, which saved an another 10 lbs over the generator used in 1962. The bore, stroke and horsepower (325) were the same as the 62 390 however this was a lighter, more agile 390 than the old design.