The band had added layers of overdubbed vocals and haunting instrumentalism to the song.
Godley was whispering the line "Be quiet - big boys don't cry" in the control room when Cathy popped in to deliver a message.
The then 19-year-old had only been working at the studios for about a year and did not want to disturb the band.
However, she knew she had to pass on a message.
She recalled: "I could see they were very busy, so I just went to the door and whispered to Eric: 'There's a phone call for you.'
"They just said 'That's it."
Impressed by the tone of her spoken voice, the band immediately asked her to replace Godley in committing them to tape.
"I immediately said no way," said Cathy, who is now known by her married name of Warren.
She said she remembered thinking: "I can't sing."