It's up to you girl. I'm going to ask a question here though:
Is there any positive contributions this girl has made to your friendship? Is there anything about her that makes you happy? Or something she does that shows that she likes you/cares about you as a friend?
some people have moments at which they are really unpleasant people. Those same people sometimes have moments in the which they are really great people.
You have been willing to overlook her actions in the past. that's fine. if you choose to do so again, that's your choice, and your emotions which you are choosing to put on the line. But if the relationship is entirely one-sided, if what you have described describes everything, the relationship will likely deteriorate to a point that it is unrecoverable.
friendship is a two way street. one person might do more walking than the other, but if only one person walks, than it's a terrible friendship.
if this is one of those terrible friendships, I will offer a third option: you can become more of an acquaintance than a friend. Maybe someday you might become close friends again.
I would say you should sever your friendship (if that's what you call it) with this person. It sounds like they're not worthy to have friends if they treat others like something dirty on the bottom of their shoe.
Ignoring someone is petty and manipulative, but if it gets results (in this case, you apologising and grovelling for forgiveness), she'll keep doing it. Don't reward her childish behaviour; let her get on with it and then she can talk to you if she stops sulking. Don't try to apologise to her, initiate conversation or anything until she's ready to grow up. It's all very well to be annoyed - who doesn't get irritated at friends from time to time? - but past the age of eight nobody should think that the silent treatment is the best way to solve conflict.