65 Minute Layover in Dublin - Is That Enough Time?

Is this one through ticket or two separate tickets?

• If it is one ticket, the airline would not have accepted the booking if they thought this was not possible. Your luggage will be labeled through to Rome. You will stay "airside" and not go through immigration in Dublin. If your first flight is late, it is the airline's problem to get you on a later flight at their cost. That said, it still sounds tight to me, Transatlantic flights can easily be 1 hour late landing.
• If this is two separate tickets, NO HOPE. You would have to go through immigration, pick up your bags, and check in again for the Rome flight, allowing for the minimum check-in for that flight. Even without bags, and your second boarding card printed out before you left home, you still need to get to the gate before that closes (typically 30 minutes before take-off) - NO WAY!

It is all on one ticket through Aer Lingus.

you must be coming in from Chicago; i've taken that route; it is enough time
the worse is that they will put you on the next flight to FCO if your flight is late coming in

We've taken that flight from Chicago twice. If it's on time, 65 minutes is plenty. It's an easy connection. It looks like Aer Lingus may have two flights a day to Rome, so if your Chicago arrival is late, they'd probably put you on the afternoon flight.