Does anyone know the best way to get to Rome from Dubrovnik?
And the likely cost of thyis.

Many thanks,
Daniel.

The only way I know of is to fly from Dubrovnik Airport to Rome Fuimancino with Croatia Airlines.

From the timetable it appears they have 3 flights a week. One on a Tuesday, one on a Sunday (both direct) and one on a Friday, where you have to change at Split.

You can contact them at http://www.croatiaairlines.com

Sorry, don't know the cost, you will need to contact them.

You could try taking the ferry to Ancona or Bari and then a Eurostar to Rome.  The ferry price should be roughly €50 and train about the same.  Sounds expensive but it might still be the cheapest option... unless you want to spend 3 days on buses.