In 1959, at the age of thirteen, Anne-Marie first met her future husband, her triple third cousin, Crown Prince Constantine of Greece, Prince of Denmark, who accompanied his parents, King Paul of Greece and Queen Frederika, on a state visit to Denmark. They met a second time in Denmark in 1961, when Constantine declared to his parents his intention to marry Anne-Marie. They met again in Athens in May 1962 at the marriage of Constantine's sister Sophia to Prince Juan Carlos of Spain and in 1963 at the centenary celebrations of the Greek monarchy.
In July 1964, the announcement of their engagement raised the polite protests of the Left in Denmark. Anne-Marie and Constantine were married on 18 September 1964 (two weeks after Anne-Marie's 18th birthday) in the Metropolis, the Greek Orthodox Cathedral of Athens. As Queen of Greece, Anne-Marie spent much of her time working for a charitable foundation known as "Her Majesty's Fund" which provided assistance to people in rural areas of Greece.