I am traveling from Mexico City to Rome, changing planes in Frankfort. If I go through passport control and immigration in Frankfort, does that mean that I will not have to go through either once I arrive in Rome?

When you arrive at Frankfurt, your gate will be in non-Schengen area of the terminal. To get to the gate for your flight to Italy, you first have to go through passport control to enter the Schengen area. When you land in Italy, there will be no further checks.
Where you go through customs depends if your luggage is through-checked to Italy. If so, you theoretically go through customs when you land in Italy, but most likely just walk through. If not, you have to collect your bags at Frankfurt, go through customs and then check in for your next flight.

If your bags are through-checked, they will carry distinctive tags that will make it clear in Rome that your bags originated outside the EU, and are therefore liable to be checked. In practice you'll just walk through, as Alec says.

But I'm puzzled at your question. If you're changing planes in Frankfort, you'll obviously have to go through immigration in Rome, as Kentucky isn't in Schengen. And I'm not sure there are direct flights to Rome anyway.

Isn't Frankfort the place where a Delta flight landed ten years or so back thinking it was Louisville?

I strongly recommend getting your booking changed to Frankfurt.

Interesting info since I've always wondered why nobody checks my passport when I arrive in Frankfurt from the US to connect to Basel, CH/Mulhouse, FR airport. In Basel nobody has ever bothered to check my passport because as far as they know I'm coming from Frankfurt, or may I just look so innocent....

Flanner: a travel agent you AREN'T (Thank God)..stick to the questions at hand rather than your warped geography and heresay, OK?

Alec has answered it the original question. Immigration in Frankfurt, customs in Italy.

As for ingrid's question about Basel, I don't know for sure, but Switzerland is not Schengen, so perhaps that flight is considered an "international", even though some passengers will exit to France or Germany instead. Just a wild guess...

I'd like to thank flanneruk for such an informative response. It's added greatly to my travel plans.