Guto Bebb, a Welsh Conservative MP and minister for defence procurement, resigned on Monday night ahead of the vote.
Glen Davies, Tory MP for Montgomeryshire, commented on Mr Bebb’s resignation, saying: “I am very upset that Guto has felt the need to resign from the Government.
“He is a good friend and was an outstanding minister.
“Guto is a man of deep principle.
“He is such a talented politician that I cannot believe he will not find another important role in the future.”
Mrs May narrowly saw off defeat in the amendment after MPs voted 305 to 302 in support of the amendment.
She was nevertheless accused of bowing to Brexiteer demands after agreeing to a series of amendments put forward by Jacob Rees-Mogg, Chairman of the European Research Group.