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For child free ships your best option is probably P&O as they have several child free ships- you will need to check out however what is in Southampton. I would not discount Princess or NCL however if you don't mind a more american approach to life. Depending on your budget may be Qe2 may be an option too.
P&O arcadia does the Med from Southampton next June and I believe that is an adult only shi.
P&O Cruises have two ships which sail from Southampton, and are child free, Artemis and Arcadia.
Artemis choices as follows: June 6th (14 nights), Artic Explorer, a cruise all way from Stavanger to Honningsvaag calling at numerous ports between. June 20th (14 nights), Northern Capitals, taking in Norway, Sweden, Finland, Russia, Latvia and Denmark. July 4th (16 nights), Iceland and the Fjords, Iceland and the Southern fjords of Norway. July 20th (14 nights), Baltic Delights, taking in Norway, Denmark, Sweden, Finland, Russia, Germany.
Arcadia choices are: June 7th (14 nights), The Dolce Vita Cruise, Western Mediterranean, Barcelona, Monte Carlo, Florence/Pisa, Santa Margherita, Rome, Alicante, Gibraltar. June 21st (14 nights) Land of the Vikings, Norway plus Spitsbergen. July 5th (16 nights) Truly Mediterranean, Adriatic, Malaga, Katakolon, Dubrovnik, Venice, Korcula, Corfu, Palma. July 21st (13 nights) Top of the World Cruise, a cruise all way from Bergen to Honningsvaag calling at numerous ports between.
If the above has whetted your appetite, try going to the P&O website at www.pocruises.com and request a brochure, then to get a good deal try one of the Cruise Travel Agents (I use Liverpool Cruise Club), they always give a better deal than going direct through P&O.
Best of luck