There is such a bus, which runs frequently. It does not leave from in front of the train station where the city buses congregate, but rather, from the bus station to the right of the train station as you face the latter. I'm happy to have scoped this out today, as otherwise I would have had a bit of a panicky departure morning tomorrow. Addendum: have now taken that bus; it does stop briefly at the Piazzale Roma, then goes straight out to the airport.
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