The piece was originally intended for Arwen's vision of Eldarion (her future son) and Aragorn. But instead, Renee Fleming's Evening Star was used.

Sissel recorded a song that was intended for the Houses of Healing which was bumped to find a spot for the orphaned Arwen's Song. (Sissel's song can be heard toward the end of the ROTK EE Fan Credits and on the CR-ROTK, The Last Debate.) I suspect this was a favor for Liv Tyler. HS comments on this decision, as does others. They are gracious about this decision but I think it was a mistake. To me, Arwen's Song is jarring. It uses English making this, and the singing heard on the banks of the Bruinen, the only non-Hobbit sung English songs. There's also something about the style of the orchestration and the singing that make it stand out - again... in a jarring way, not a pleasant way. I think this is a song that one either loves or hates.

The final credits list featured artists and song credits. The differences between the TE and the EE are pertinent to this song. For 'Featured Artists', the EE lists Liv Tyler and Sissel and the song, "A Shadow Lies Between Us". I spent a lot of time figuring out which song, Liv's or Sissel's was "A Shadow Lies Between Us." Since "A Shadow Lies Between Us" falls (in the credits) between "Edge of Night" and "Aragorn's Coronation", (and the songs seem to be listed in order of appearance) that might mean it referred to Liv's Song in the Houses of Healing rather than Sissel's song in the Fan Credits... unless Sissel's song was in the Houses of Healing when the credits were prepared and then changed at the last minute. But why would they list both artists but not both songs?

We knew that Sissel's song drew some of it's lyrics from the source text used in the Breath of Life, which was entitled in that sheet music, "Aragorn". The Annotated Score of the TTT called that source text, "The Grace of the Valar also known as The Breath of Life". It also gave us some additional lines for the source text including one that read (in it's English translation), "Shadow Lies Between Us". Ahh... so that must mean that the ROTK credits Sissel's song and not Liv's.

But Doug Adam's had this to say:

You should... understand that Arwen's Song went through several lyrics... some of which went on to be used in other compositions instead, some of which went totally unused.

Perhaps some of the original lyrics of Arwen's/Liv's Song went on to be used in Grace of the Valar/Breath of Life and by listing "A Shadow Lies Between Us" in the credits, they were referring to both songs. It doesn't quite explain the change in names. But the names of musical and vocal pieces seem a bit mutable. We already know the lyrics to the Breath of Life song were called in the sheet music, "Aragorn", and in the AS-TTT, "Grace of the Valar/Breath of Life". And the song listed in the ROTK credits as, "Aragorn's Coronation", was titled, "Elessar's Oath" in the AS-FOTR. Doug Adams said this:

Sometimes a text would get one title, but the composition in which it was used would get another. You’ll even notice in the FOTR: CR credits there will be some “also known as” credits regarding texts and such. Most this was due to the legal filing of the cue sheets and copyrights. For example, the text would have its title (as decided by Philippa or Fran or Tolkien), the composition would have one title, it would get another title when edited into the OST, then another title again as it was entered into the CR. So there are a lot of multiple titles out there. (Sometimes there were even alternate text titles to deal with as well!)

If "Grace of the Valar/Breath of Life" drew material from "Arwen's Song", then we have another little mystery/magpie curiosity to examine. The only other non-Hobbit sung English piece is heard on the banks of the Bruinen as Arwen pleads for Frodo's life. The source text for that song was given the name (in the AS-FOTR) as "Arwen's Prayer". But the words are:

What grace is given me

Let it pass to him

Let him be spared

Mighty Valar

Save him.