Both flights arrive and leave at Terminal 1
As already mentioned, you will go through immigration in Frankfurt.
Just so you know, Terminal 1 consists of three separate buildings. The first building contains Concourses A and Z. Concourse A is for domestic flights (between Schengen countries). Concourse Z is on the floors above Concourse A and is for flights to/from airports that are not in Schengen countries. If you are fortunate and your flight from the US arrives at a Z gate and your flight to Rome departs from an A gate, you will just need to go through immigration while going downstairs (escalator) to the A Concourse.
However, if one of our flights arrives or departs from a gate in the B Concourse and the other one uses an A or Z gate, you will have to change buildings. The fastest way to do that is probably to use the pedestrian tunnel between the buildings, but that will probably entail security when entering the new concourse. (It might also be possible to use the SkyLine shuttle between the buildings, but that will still entail going through security when entering the new building.)
If you know your flight nubers, you can check the gates for arrival and departure on the Frankfurt airport website.