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 $ tail -f /path/to/file.log | sed '/^Finished: SUCCESS$/ q'

Aug. 22, 2012, 8:29 a.m.Janos

Explanation

tail -f until this exact line is seen:

Finished: SUCCESS

The exit condition does not have to be an exact line, it could just well be a simple pattern:

... | sed '/Finished/ q'