In 1883 Constatin Brun received a Master of Laws degree from the University of Copenhagen.
In 1885 Constantin Brun was an assistant in the Ministry of Foreign Affairs. In 1887 he was a Secretary of Legation, Berlin in 1887 and in Paris in 1891. From 1895 to 1906 he was a minister of Denmark in Washington, District of Columbia, United States, and in London, from 1908 to 1912, in Washington in 1912. In 1907 Brun was a delegate from Denmark at Peace Conference in The Hague.
In 1916 he signed Treaty ceding Danish West Indies to United States Army in New York. In 1897 he was a chamberlain to the King of Denmark.