He served in the ministries of Sir Robert Menzies, Harold Holt, John McEwen, John Gorton and William McMahon. In March 1964 Menzies appointed him Attorney-General. In that capacity he played a significant role in the 1967 constitutional referendum affecting the status of Aboriginals. On 7 April 1965, the Menzies Cabinet decided that it would seek to repeal Section 127 of the Constitution, which excluded indigenous people from the population count, but made no firm plans or timetable for such action. In August 1965, Snedden proposed to Cabinet that abolition of Section 127 was inappropriate unless Section 51(xxvi) was simultaneously amended to remove the words "other than the aboriginal race in any state". He was rebuffed, but he gained agreement when he made a similar submission to the Holt Cabinet in 1966. The referendum went ahead on 27 May 1967, and was resoundingly approved.