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The weather could be cool and damp, sunny and warm, or anything in between. Dress accordingly. (You'll probably be okay with Dockers and sports shirts, a long-sleeved shirt or two, and a water-resistant windbreaker or golf jacket in case it drizzles.)
I really like the Ex Officio "Air Strip" men's shirt that Magellan's sells, and which I reviewed at:
It's comfortable in all kinds of weather, the sleeves can be rolled up and buttoned in place with tabs, and it drip-dries in a couple of hours when washed by hand. Recently I bought a similar (and cheaper) REI Sahara Tech shirt (www.rei.com) that I like even better because it looks a bit dressier and could be worn with a sport jacket in a pinch.
As for women's clothes, I'll leave that topic to the female members of the forum. :-)
I'd guess that most people do their clothes shopping before they board the ship, except for logo cruisewear and emergency items. (Mind you, I don't claim to know anything about the retail business--I don't even like to shop.)
I'm on the same cruise June 14 from Stockholm to Dover. I checked all the ports we'll be visiting on an Internet weather site and it looks like the average high for all ports is in the 64 - 70 range and the average low 49 - 52. I did almost the same cruise 4 years ago mid to late May and the temps were quite chilly. It also rained in almost every port so be sure to bring some kind of rain gear. I bought a fold up plastic raincoat at our local AAA store and it more than serves the purpose--a little sturdier than the "cheapies" at KMart, but still convenient to pack and it has a hood. Umbrellas are a real nuisance IMHO. I plan to take a few pairs of khaki's, a couple of sweaters with some short and long sleeve tops. Hope this helps.
We were on the same cruise last year the end of July and beginning of August. It was the hottest summer they ever had in the Baltics. I would wait until a few weeks before you go to guess what kind of weather you are going to have. I was packing for cool and sunny weather that we were told to plan for when my daughter called me and said it was 92 in Helsinki. I threw in a few lighter weight things and naturally those were the only things worn on the trip. Those countries are not prepared for heat. No air conditioning.