I really don't know, since I'm exactly like your friend, I'm always putting myself down because thats just how I think and it's how people have trated me in the past, consider it habit now. Maybe you need to have a real sit-down and talk with her and some close friends, maybe you just need to leave her alone for a bit. At least tell her how you feel so she knows why you might brush her off or not hang out as much, otherwise she might read it as you just don't like her, and that will only make things worse for her. I know thats how I would want my friend to react.
Good luck!

Let your friend know how much you care about her and that you are worried about her or that you want to help her over come these negative comments. Most people who talk that way don't care much about themselves and they are looking for attention. Let her know that its hard for you to hear her talk that way and find out what you can do to help. Maybe she just needs someone to acknowledge her and let her know that they are listening. Good luck!

I would write her a letter of everything you love about her, then say it hurts you to hear these things because they aren't true or every time she says something negative say something positive about her. Don't brush her off that'll only make her fall more and think more negatively about herself. Show her love and compassion she needs that from you.

We all do sometimes! I know that if we just say something positive after a negative along with a loving letter, that person may start seeing themselves as we see them. It works with ourselves as well everytime you say something negative write something positive about yourself.

avoiding her is the last thing which you should be doing right now
try to make her realize how good she is by telling her about the great qualities which she has
avoiding her would just make the matters worse , she might take that reaction as a confirmation of her insecurities about herself.. might just worsen up things
and when you do appreciate her please be genuine because she would come to know if you are making up stuff
and like Chelsea said she might just be hoping that you would talk to her about all this and help her to get out of it...
all the best.. i hope everything gets fine ... :)
and by the way i appreciate what your doing ....
god bless :D

I have a friend who is exactly like that.

At first, I started trying to put her doubts to rest - whenever she said something nasty about herself, I'd always reply saying something wonderful. Eventually, she started calling me a liar, so I decided to try some tough love. I told her I have my own self-image problems to deal with and I was only telling her one more time that she was beautiful/funny/smart and that was it. I did this a couple of times, even suggesting she should try counselling, (making her cry in the process) and then after a week, she said it again and I just said to her "Every time you talk about yourself like that, I'm going to ignore it because you don't listen to what I say, so stop it." And I did. Everytime she said she was ugly or fat, I'd act as if she said nothing.

And she stopped. That worked for me. And she seems to be a happier person for it, although I'm not really very sure.

Your friend has a very low self esteem and she needs support, she needs a good friend. I agree with the other girls that post you don't want to avoid her it will not help her one bit. The letter advice is good. or maybe try gathering a bunch of your common friends and get a card writing , taking turns on what everyone likes about her and how they feel about her. I, myself have a very low self esteem but i have a good friend support system that is helping me build it back. Another advice which you might have to be-careful on how you approach her with is going for counseling . I go for counseling and it really has help me get to the bottom of why i feel so down about myself all the time. hope it helps!