Galatians 2:20: "I have been crucified with Christ. It is no longer I who live, but Christ who lives in me. And the life I now live in the flesh I live by faith in the Son of God, who loved me and gave himself for me."
Galatians 6:14: "But far be it from me to boast except in the cross of our Lord Jesus Christ, by which the world has been crucified to me, and I to the world."
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The text is public domain:
In the cross of Christ I glory,
Tow'ring o'er the wrecks of time.
All the light of sacred story
Gathers round its head sublime.
When the woes of life o'ertake me,
Hopes deceive, and fears annoy,
Never shall the cross forsake me;
Lo, it glows with peace and joy.
When the sun of bliss is beaming
Light and love upon my way,
From the cross the radiance streaming
Adds more luster to the day.
Bane and blessing, pain and pleasure
By the cross are sanctified;
Peace is there that knows no measure,
Joys that through all time abide.
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Galatians 6:14 seems to be reworkt into the title line. Rather than "But far be it from me to boast except in the cross of our Lord Jesus Christ," the hymn has "In the cross of Christ I glory." Both are essentially the same view, just from different directions.
I don't think there are specific phrases in the hymn that come from Galatians 2:20, but the hymn - especially the last three verses - does seem to describe the "peace and joy" that comes with knowing what that verse expresses.