Ellicott is one of only six politicians to have served in both the Parliament of Australia and the Federal Court of Australia, along with Nigel Bowen, Merv Everett, Tony Whitlam, John Reeves and Duncan Kerr. Ellicott was admitted to the New South Wales Bar in 1950 and was Solicitor-General of Australia from 1969 to 1973. He was elected as the Liberal member for the Division of Wentworth in the 1974 election.
He was Attorney-General in the Fraser Ministry from 1975 to 1977.
Ellicott resigned as Attorney-General as a result of a dispute with Malcolm Fraser over the payment of costs in the Sankey v Whitlam and Others case, where he believed that the Commonwealth should have paid the costs of the private individual, Danny Sankey, as well as that of the politicians, Gough Whitlam, Rex Connor, Jim Cairns and Lionel Murphy, but Fraser disagreed. Ellicott was reappointed in the third Fraser Ministry (1977 to 1980) as Minister for Home Affairs and Minister for Capital Territory.
He was Minister for Home Affairs and the Environment from November 1980 until his resignation on 17 February 1981 to become a judge on the Federal Court of Australia. Ellicott resigned from the court in February 1983.
He is currently an arbitrator on the Court of Arbitration for Sport.
On 20 November 2007, he was named as chair of the tribunal to investigate allegations of misbehaviour against the suspended Chief Justice of Fiji, Daniel Fatiaki. In May 2006, the Australian Olympic Committee awarded him the Olympic Order of Merit, particularly in his role of establishing the Australian Institute of Sport when Minister for Home Affairs.