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The hours of the 24-hour clock starting at midnight until, generally, dawn or, more specifically, 5-6 am.
I will be in BTC until the wee hours of the morning.

We partied through the wee hours of the morning, and passed out after dawn.
by RW6 September 26, 2005
the time spent in your life peeing
i wasted my life in the wee hours
by lordgorlok October 13, 2006