But I don't think she gave up her immortality as such; presumably her spirit would return to the Halls of Mandos for reincarnation just as that of any elf who died in Middle Earth would.
No - to quote Elrond when he first finds out that Aragorn is in love with her:

'But there will be no choice before Arwen, my beloved, unless you, Aragorn, Arathron's son, come between us and bring one of us, you or me, to a bitter parting beyond the end of the world.'

Actually, I think this is one of the ways that it works much better than B/A - Angel wants to be human (especially since otherwise he's doomed to hell etc), whereas Arwen has to give up her kind and an immortal life that is only good.