We chose the Gradska Kavana ("City coffeehouse") for our stop after a few hours of walking through Dubrovnik streets. It's located in the very centre - at the end of Stradun (the main boulevard). It sits just between the Arsenal and Town Hall - almost impossible to miss.
Inside, the rooms were elegant yet somewhat rustic - reminding us in a way of Vienna cafes - no wonder, as the city was ruled by Austrians in the 19th century - matched by the staff's elegant attire - but we chose to sit on the terrace and have a look at the Stradun teeming with life in the early evening. There's also another terrace, with a great view of the port and the city walls.
The coffees we had were excellent - from the very Viennese coffee with whipped cream, to the quintessential Mediterranean espresso. One of the ladies had a cup of hot chocolate - it was also great, like a hot chocolate mousse. We didn't get any cakes - but the selection was quite wide and they were looking delicious.
The prices were moderate - a bit above other cafes, but still quite acceptable - the coffees were mostly around 10Kn.