1) Is it knowing that she does not like you which interferes with your work?
or
2) Is it because she does not like you - and she is doing something to purposely interfere with your work (ex. not properly doing her part in shared assignments with you; unwilling to co-operate with you in work-related duties etc.)?
From the threads and posts I've come across from you - the vibe I get is that it is #1.
If I were in your position and HAD to ask (normally I wouldn't; I simply move on if someone does not like me); I would state more along the lines of: "Hi Jennifer. I hope you do not mind me bringing this up, but I have been feeling that you dislike me - and if so - I would like to know why. I only ask this because as co-workers, I feel that it is important for everyone to get along as we all want to be part of a positive work environment."
Personally, I would NOT elaborate that I am unable to function as a working adult only because of her dislike towards me. (Ex. "Jennifer, when you didn't like me from day 1, I had a hard time getting along with others and getting my work done. When I do get it done, it wasn't good quality."). Nor would I ask "Will you be my friend?" at the end of it all because it seems very desperate especially if she has made it clear that she does not like you and is not interested in your friendship already.
Sorry, but you should be able to still function as a worker and be able to do your job even if just one co-worker does not like you - and is not doing anything to purposely make you not work. If she does want you as a friend - you will see by her actions and there is no need for you to keep asking her.
From her response, she may or may not tell you the truth. In fact, she may dislike you but have no reason to justify it!
If that is the case, I would like to reiterate that sometimes people just won't like you for no reason - and I'm not speaking just to you, but I mean in general terms for everyone.
You could be nice to them, buy them gifts, help them out - and they will still not like you! Of course, trying to "ask", "bribe" or "buy" your way into friendship has no rewards. Do not put forth effort into someone who does not do the same to you.
(Ex. I had one co-worker - who I would greet in the mornings when passing by - and she would roll her eyes at me and ignored me for probably a year before she quit. I remained professional - didn't let it affect me - and continued working productively.)
Again, if it is #1 - the issue is with you. If she doesn't like you - she doesn't like you. If you keep on over-analyzing constantly to the point where it interferes with many aspects of your life; you need to find help because it is not healthy and scarily obsessive.
***
If it is #2 - where she is purposely causing trouble to you such as (Ex. Lying to the supervisor and saying you did things that you did not do/ screwing things up before passing it to you etc), then you can confront her and also go to the supervisor if things do not improve. You can mention to the supervisor that you brought it to Jennifer's attention and attempted to solve it among yourselves before bringing it to a higher level. If your supervisor does not work out a solution/ ignores you etc; best to find another employer.
That's why it's important to be a good worker regardless. You need to build a good reputation and credibility. If you let every little thing get to you and let it cause trouble for you - even the most patient of supervisors - your words will not hold much value. Again, in this type of case, it's best to start fresh - and work better - with another employer.
***
Whatever Jennifer's response is - be patient, and do not let yourself get sad, mad etc especially in front of her face or your co-workers. You need to be controlled and professional especially when it comes to confrontation at the workplace. Do not rage out under any circumstances. Even if you're in the right, it makes you look wrong.
I hope it works out for you because it seems to be a big issue for you.
I do not mean any offense in my post...and I hope that you at least take the time to read it since I am really trying to help you.
