RUN.
IN.
PARALLEL.
WITH.
OLD.
SYSTEM.
DUPLICATING.
THE.
REAL.
LIVE.
DATA.
Anything else is not "testing", it's just guessing if it will work or not.
Branch off the real data (or even just a portion), feed it through your test system, check that the answers match and/or that the behaviour is as expected.
Leave it like that for a significant period of time. Only THEN do you think about actually swapping out the old.
Biting the hand that feeds IT
