Great example, BarensMom, I do understand one reason the OP might differentiate it from hers - you "quit" immediately on learning your ride-cadger had a readily available option other than you, but OP has known that her ride-cadger has a car/ride available for some time.
OP, it's obviously your choice on timing and how you address your situation. You may want to think *now* about whether it would be easier *for you* to have the conversation with your rider this week, giving her a few days to plan for a ride other than you (she *has* one, just has chosen to not use it) - and then avoid, or limit the time you have to spend with her in a car, or having the discussion while sharing a ride.
While I would generally not advocate ending a "personal relationship" over the phone, that is not what you would be doing. You would be advising a co-worker that you are not continuing a carpool, and you know she has other transportation available, just want to give her a few days notice before she starts her new contract so she knows you won't be driving her.