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Now, you're still not guaranteed to produce a Unicorn foal when you breed at 6.23(am/pm), but if you don't breed at these times, you will definitely never produce a Unicorn foal.
It's believed that you have a one in six chance of producing a Unicorn foal, so be patient. However, someone I know bred their Unicorn 20 times and never produced a Unicorn foal!
UPDATE: As of 20 April 2011, you now have a 1 in 5 chance.
So good luck. You might need it.
Patsy