1) If you mean first Bat Girl (before PRe-CRisis) Betty Kane (the niece of the Bat Woman), then the Black Widow wins with amazing ease.
2) If you mean the first Bat Girl (after Pre-Crisis), Barbara Gordon, then there is a big chance that the Black Widow will lose. But it is not written in stone.
3) If you are referring to when the Huntress (Helen Bertinelli) was Batgirl (during No Man's Land) then it is more or less like the Barbara situation. I'd say the Huntress-Batgirl would win, but again it is not written in stone.
4) But if you are talking about the CURRENT Batgirl (Cassandra Cain), then what will happen is that this Batgirl will thrash the Black Widow so badly that it will not even be funny. Cassandra Cain is simply beyond anything the Black Widow can bring to the fight.
IT shouldn't even be called a fight. Not even a beating. IT is simply put the same as a parent taking an iron rod and whipping a 3 month old baby.
Basically illegal!
What Batgirl would do to the Black Widow would be a crime!