Hi happyfeet, thinking just for a minute (!!) that you don't want to leave them. Could you get together the proof that they're lying to you and discuss the lies you've found with them (in a non-confrontational way). Then if they can be open with you and you can resolve the issues behind why they are lying between you then maybe there's a chance.
There maybe some workable reasons for their lying which once out in the open you can work on e.g. they may be scared of losing you if they were to tell the truth, they maybe ashamed of something, they may need emotional support in some way but didn't want to "burden" you, they may have had negative experiences in the past from telling the truth, they may have been brought up/or had influences in their life whereas "you just don't talk about certain things", they may be unsure of the relationship you have................but maybe you both can still keep a good relationship with them telling the truth.
But you know if it can't be resolved, I've got to say that the only option has to be to end the relationship. I know it may well hurt really badly, but if you compare it to continuing the relationship....the lack of respect you're getting.......the inability to really trust the person......the not knowing..........the wondering........the hurt it's going to be causing you.......
And of course the "love" aspect, maybe you're loving the person you want them to be (who they are without the lying) as opposed to the real person right in front of you. And (if it can't be resolved) then there's no getting away from who you're loving just isn't actually there.
Hopefully you two can work things out with some more communication. Just remember they may be a bit on the defensive when you raise it with them, so you might just need to finish it with making sure they know how you feel and raise it again another time (just don't leave it too long before raising it again!!).
Best wishes
Alison