He was then educated at the prestigious Lycée Henri-IV in Paris, before studying at the Institut d'Études Politiques de Paris, better known as Sciences Po, graduating with an undergraduate diploma in 1961, Master of Laws in 1962 and Diploma of Superior Studies in Public Law in 1964.

In June 1966, Costa was appointed Auditeur in the Council of State, a body of the French national government that provides the executive branch with legal advice and acts as the administrative court of last resort.

From 1968 to 1973, he lectured at Sciences Po, and from 1981 to 1984 was Director of the Office of the Minister of National Education, Alain Savary.

From 1985 to 1986, he led the French delegation negotiating the construction of the Channel Tunnel, and from 1985 to 1989 taught at the International Institute of Public Administration.

On 1 January, he was appointed the judge in respect of France at the newly-permanent European Court of Human Rights.