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Image slightly changed from Wikipedia
This is the Website of the Port Sochi with timetable:
http://www.morport-sochi.ru/main.php?nav=lines&lang=en
The ferry to Batumi via Poti goes all year around. Twice a week, but not always on the same weekdays.
Call some days in advance to Sochi:
007 8622 609603
Today they told me a price of approx. USD 60.00
By the way, I am reading a great book on the History of the Black Sea by Charles King:
The Black Sea A History
Here is a link to Amazon