Free-to-use (ie public domain or copyright-free) hymns can be copied and sung to at least one tune with no concerns about copyright or performance licensing - sometimes in a limited setting. The number of these hymns which are suitable for Christian unity services is limited, so a number of still-copyright hymns are are also listed.
Seasonal noteThe Week of Prayer for Christian Unity is one of a number of days and weeks in which people are asked to pray for particular needs. It has been observed since 1894, initially by the Church of England but joined by the Roman Catholic Church in 1895 following a worldwide request of Pope Leo XIII. Leo established it as a novena “in perpetuity" in 1897 (in "Divinum illud munus”).
Since 1908 the octave has been observed in January dates starting on the date of the (now former) feast of the Confession (or Chair) of St. Peter and ending on the Feast of the Conversion St. Paul.
The Catholic lectionary has suggested readings for Masses for Various Needs and Occasions, one of which is Unity of Christians, and these are given below. The hymn suggestions are based on these readings, and also on the general themes of Christian unity.
Psalm 22 - The Lord is my shepherd there is nothing I shall want / He guides me along the right path / You have prepared a banquet for me / Surely goodness and kindness shall follow me.
Second Ephesians 4:30 - 5:2 - Do not grieve the Holy Spirit ... who has marked you with his seal ... Be friends with one another, and kind, forgiving each other as God forgave you in Christ..
Gospel John 17:20-26 - Jesus prays for the disciples, and for those who will believe in him through the words of the disciples ... May they all be one in us as you are in me and I am in you.
Free-use hymn suggestionsAt Thy First Eucharist
Beloved Let Us Love
Hail to the Lord's Anointed
In Christ There is no East or West
Let us Break Bread Together on our Knees
Of the Father's Love Begotten
The Church's One Foundation
The Lord's my Shepherd
We are Gathering Together Unto Him
We are Many we are One
Your Hand O Lord has Guided
Suggested hymns that are still copyrightAll Are Welcome - Haugen
And They'll Know We Are Christians - Scholtes
Bind Us Together - Teach Me Lord to Wait - Gillman
Companions on the Journey - Carey Landry
Diverse in Culture, Nation, Race - Duck
For the Healing of the Nations - Kaan
Forth In the Peace of Christ We Go - Quinn
In Christ There is a Table Set for All - Stamps
Make Us One - One Communion of Love - Marchionda
O Christ the Great Foundation - Lew
One Spirit, One Church - Quinlivan/Keil
Seek O Seek the Lord - McAuley
Ubi caritas - Taizé
We Are Many Parts - Haugen
Suggestions about hymns that could be included are welcome: please leave a comment below.