The New York Times has always called it "the Concorde" in text, but may abbreviate to "Concorde" in headlines.

Headline example, October 20, 1963:

C.A.B. Head Assays Supersonics And Finds the Prospects Gloomy; Tells Senators Concorde is 'a Loser' and Warns That American Airlines Probably Will Fail to Profit From New Planes

Text example, January 30, 1964:

The Concorde is the supersonic airliner being built jointly by the British and the French.

In this article, both headline and text, November 13, 1977:

Aloft at Mach 2, a Critic Rides and Rates the Concorde; Riding and Rating the Concorde

Before I begin this diary of a flight on the Concorde, I must, as an obedient child of the 1960's, explain why I decided to do such a dastardly thing in the first place.