At this point Arwen has already given up her immortality, but now she grants the grace of the Eldar to Frodo, not giving him immortality since he is not one of the elves, but instead granting him a home in the undying lands when he does pass.
So, No she did not give it to Frodo. Immortality could be given up, but not passed to another. She gave up her immortality to marry Aragorn, who was now the king of both Gondor, and Arnor (The reunited kingdom).
Arwen made a sacrifice, but it was not for the sake of Frodo, but for her own sake and her love of Aragorn.
Did Arwen lose her immortality? The answer is yes.
So there we have it, definitive proof from Tolkein himself that Frodo, and his other mortal counterparts, did eventually perish in The Undying Lands.
Since Frodo saved all of Middle Earth, he is granted access on the last ship to leave Middle Earth in place of Arwen, never to return.
Aman was known somewhat misleadingly as "the Undying Lands", but the land itself does not cause mortals to live forever. However, only immortal beings were generally allowed to reside there.
Arwen reciprocated Aragorn's love, and on the mound of Cerin Amroth they committed themselves to marrying each other. In making that choice, Arwen gave up the Elvish immortality available to her as a daughter of Elrond, and agreed to remain in Middle-earth instead of travelling to the Undying Lands.
And after their marriage, both Aragorn and Arwen are mortal. Arwen, therefore, could not set foot in the Undying Lands without special permission. And she wouldn't be likely to go after it in the end. So unfortunately, she would not go there, unlike Legolas, after Aragorn's passing.
However, he does not possess the ability to live forever, and he cannot choose to be mortal or immortal like Arwen, the half-Elf he loves and eventually claims as his wife. Because Aragorn is mortal and Arwen chose to remain in Gondor and become mortal, their son, Eldarion, is mortal as well, despite his Elven lineage.
According to 'The Tale of Aragorn and Arwen', from Appendix A of LotR, Aragorn II (Elessar) and Arwen had at least three children: one son, Eldarion, who succeeded his father as king, and at least two daughters, as there is a mention of unnamed 'daughters'.
At this point Arwen has already given up her immortality, but now she grants the grace of the Eldar to Frodo, not giving him immortality since he is not one of the elves, but instead granting him a home in the undying lands when he does pass.
Tolkien strongly suggests that it was her marriage to Aragorn that sealed her fate and rendered her mortal. Marriage was hugely significant in Tolkien's works. At that point, she became long-lived but mortal. However, after Aragorn's death, she simply chose when she wanted to die -- as did he.
Originally, Tolkien intended for Éowyn to marry Aragorn. Later, however, he decided against it because Aragorn was "too old and lordly and grim". He considered making Éowyn the twin sister of Éomund, and having her die "to avenge or save Théoden".
Legolas and Arwen in Rivendell
These two elves aren't actually seen together at any point in the trilogy, but they do in fact have a strong sense of kinship, which can be seen in one of these unused clips where Legolas arrives in Rivendell and Arwen and a host of other elves are there to meet him.
All of the elves' fates are linked, in some ways, to the One Ring. This is both because the three elven rings of power will fade once the One Ring has been destroyed, and also because the destruction of Sauron will bring about the end of the Age of elves, and start a new Age of men.
Arwen didn't want Aragorn to die, because if he died so would she. She said she wasn't tired of living yet. But instead, Aragorn chose to die and Arwen lived a year longer before dying herself.
In the year 121 of the Fourth Age, after Aragorn's death, Arwen died of a broken heart at Cerin Amroth in Lórien, and was buried there one year after the death of Aragorn, to whom she had been wedded for 122 years. She was 2901 years old.
Eventually, Legolas came to Ithilien with some of his people, with his father's leave, to live out his remaining time in Middle-earth helping to restore the devastatedforests of that war-ravaged land.
The Gift of an Arwen Evenstar Necklace
Princess Arwen Evenstar gave her beautiful necklace to Aragon to pledge her eternal love to him. This special gift from an immortal elf to a mortal man symbolized Arwen's decision to forsake her immortality to be with the person she truly loves.
Legolas is an elf, so he is immortal. However, he is only immortal in the sense that he will never die of old age or sickness. He can die in battle.
Arwen was her granddaughter
Galadriel and Celeborn had a daughter named Celebrian, who married Elrond and became the mother of Arwen (as well as her brothers, Elladan and Elrohir).
Frodo and Bilbo were comfortable and well off until T.A. 3001. At this time, Bilbo threw an enormous party to celebrate his 111th birthday, and Frodo's 33rd, the date of Frodo's coming of age. At this party Bilbo gave his farewell speech, and made his long-planned "disappearance" and withdrawal from the Shire.
Frodo went to the Undying Lands at the end of the Lord of the Rings trilogy, but many wonder why did Frodo leave Middle-earth instead of settling down in the Shire like Sam. The reasons for the Elves inviting him are clear given his destruction of the One Ring and Sauron, but Frodo's reasons for accepting are less so.
He departed for the Undying Lands along with the last of the Elves - such as Elrond and Galadriel - and was accompanied by Bilbo and Frodo as well. With a goodbye to the other Hobbits, they sailed away into the West and were never heard from again.