Clarification: When I referred to the "original words to that verse of the song" in my previous explanation, I meant the lyrics of the beautiful 19th century hymn by Edward Mote, My Hope Is Built (The Solid Rock). Several verses of Hillsong’s Cornerstone, including the phrase under discussion here, come directly from that old hymn. That portion of the lyrics is not original to Hillsong.

I understand the apparent connection of the phrase, "My anchor holds within the veil," with Hebrews 6: 19-20. However, I am not convinced that this was the intention of the song's author. The original words to that verse of the song are: "When darkness veils His lovely face I rest on His unchanging grace; In every high and stormy gale My anchor holds within the veil." This was authored as two parts of one sentence, apparently linking the two uses of "veil." It would seem to indicate that the unchanging grace of Jesus Christ (the Solid Rock), is my anchor ("all my hope and stay" - verse 3) even in the dark and stormy times of life when I cannot perceive His presence. This interpretation of the lyrics does not negate the truth included in the other comments, but it does offer an alternative insight into the intended meaning.