New cylinder needed or a reaming job to cut a new chamber, don't jam them in you 357 cylinder, they shouldn't even fit in reality as Supers are slightly larger in diameter, but tolerance stacking often allows it too.
Also unless you hand load you will probably find accuracy a little underwhelming with a 357 barrel since most use 0.355 or 0.356 jacketed bullets.
Any particular reason to pick 38 super over 38/357? Are you shooting competition and reloading on the clock? It is easier to go with 38 short colt if that is the case. It's basically a 9mm luger length 38 spl in fact you can use all 9mm dies with the exception of the sizer die which needs to be a 38/357 die or anything in that family.