T., she is still young enough that you can and should "tell her what to do." But you can go about it in a way that lets her take the lead. Ask questions that let her open up but take care not to badger her too much with questions. Roleplay with her what gets said and ways she can respond.
It's worrying that she is so focused on her looks -- my daughter is in sixth grade and she and her peers are truly not that into how they look or whether someone is "fat and ugly." Is your daughter possibly seeing too much on TV ads, in magazines, etc. that would make her compare herself to some unrealistic standard of size and looks? Do thse "cute popular girls" at school focus a lot on who wears what and how much girls weigh? Ensure that your daughter isn't watching commercial TV or seeing magazines or other media that talk about women's looks. I still avoid the aisles in the grocery stores with those women's magazines that have a ton of "how to lose weight" articles listed on the cover, so my daughter won't have to see that stuff.
Someone suggested "Queen Bees and Wannabes" and I highly recommend it too. These popular girls are turning into queen bees, no question.
I also agree with those saying that she needs engaging and frequent activities outside school. Let her choose based on her own interests -- Girl Scouts, sports (not too competitive unless she's up for that), dance (I would avoid competitive dance because it focuses so much on appearance), art classes at the community center if she likes art, kids' theatre (if she doesn't want to be on stage, would she like to work backstage?), volunteering through a church if that's your thing, etc., etc.
Why are we all recommending activities? The point with activities is that they help a child find friends who become friends based on shared interests -- not on who wears what or who said what to whom. When the link is an interest and not "who's in and who's out" the child knows that she is valued for who she really is and not for her looks or her popularity this week. Your daughter can know while at school that she's not that "fat and ugly" person the queen bees say she is -- she's the girl who loves theatre, or who make great clay sculptures, or whatever really defines her true self.
Do talk to her about how real friends -- kids with real values -- do not tell you they like you one day then tell you they're sick of you the next week. She will not get that at first but needs to hear it. Ask her what it is about them that she wants to "be like" -- their clothes? Their popularity? Does she really want to be "be like them" if that means telling a kid like herself, "I like you...no, I don't like you today and I'm sick of you"? That might get her attention. (Girl Scouts has some very good programs that reinforce that girls are all valuable and must not treat each other that way.)
If she keeps on being miserable and these girls are truly queen bees you need to equip her to deal with them (mostly by ignoring them and finding her own friends who share interests and values). But if it gets really bad I would not hesitate to tell a trusted teacher that there is a powerful mean-girl clique. A good teacher will hear that and -- without identifying the child whose parent blew the whistle -- work to at least ensure that the mean girl clique can't be glued together during the school day (by separating them in classes etc.).
Finally, doesn't her school have a school counselor? In our schools, the counselors do monthly counseling lessons in every class on topics like peer pressure, bullying, etc. If your counselor does not do this as part of his or her job-- he or she should! I would go to the counselor and say that you are seeing this queen bee setup creating pressure and it's time for her to address it. That does NOT mean calling in specific kids, which will only get kids like your daughter on the bad books of the jerky girls. It means doing general counseling lessons for all students, and repeating over and over that popularity isn't about clothes or weight etc. The school can reinforce this and it can be effective.
Your daughter does indeed have to figure most of this out on her own but you CAN help without smothering her - you truly can!