In 1959, at the age of thirteen, Anne-Marie first met her future husband, her triple third cousin Constantine, Crown Prince of Greece, who accompanied his parents, King Paul and Queen Frederica, 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 Princess Sophia of Greece and Denmark to Prince Juan Carlos of Spain at which Anne-Marie was a bridesmaid: and again in 1963 at the centenary celebrations of the Greek monarchy.
On 6 March 1964, King Paul died, and Constantine succeeded him as King of the Hellenes. In July 1964, the announcement of the engagement of Constantine and Anne-Marie 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. The bride wore a Jørgen Bender design. Prior to the wedding, Anne-Marie converted from Lutheranism to the Greek Orthodox Church. Also, in view of the fact that she was marrying a foreign ruler, consent to the marriage was given on the condition that Anne-Marie renounced her succession rights to the Danish throne for herself and her descendants.
Anne-Marie and her husband Constantine are third cousins: they share King Christian IX of Denmark as patrilineal great-great-grandfather. They also share Queen Victoria of the United Kingdom of Great Britain and Ireland as a great-great-grandmother. The couple has five children: Princess Alexia, Crown Prince Pavlos, Prince Nikolaos, Princess Theodora and Prince Philippos.