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The only line I know that have 'adult only' ships are P&O. They have two ships the Artemis (older and smaller) and the Arcadia (just over a year old). Both provide excellent food, service etc and are primarily used by Brits. The Arcadia is more glitzy but both have great shows and on 'sea' days there are lots of activites should you not wish to relax. Only trouble is you may find one week is not enough. Whichever you pick I am sure you will have a great cruise, we did and are currently looking at 2008 schedules.
I agree with the answer from "Morris, Telford". It also depends how much money you want to spend. The more expensive the line the less likely you are to find children aboard. Silversea, and Crystal are almost always childfree as is Swan Helennic when they make their return, whereas the less you pay then the more likely you are to come across them. If you go for Royal Caribbean, Celebrity, and Princess I have found that I am often surprised by the number of children on board because you will only see them very occasionally, the children's programmes by these cruise lines being excellent. Another way to avoid their presence is to select a time when the school holidays are not operating. We've never had a cruise ruined by children in any way shape or form yet. Neil Dow.
If you want to sail from the UK try Fred Olsen's Black Prince, it isn't exactly child free but there are no facilities onboard for children so this would not be a problem (I only go on Childfree ships). If you want a fly cruise Thomson's Emerald is as above, but their Ship Calypso is Adult only. Another one (my favourite) is Orient Lines Marco Polo. Most small ships do not attract kids because they don't really have the facilities. It's not that I don't like kids, I just couldn't eat a whole one.