• Posters were put up around the Musselburgh Harbour area in the search for Elizabeth McVey
Her family said they were "devastated" by the news, and that she would be sadly missed.
The body was discovered by a member of the public on the beach near to the Renaissance Club in Cowdenhill Drive at around 10.15am yesterday.
Elizabeth had not made contact with family or friends since she was last seen in her Musselburgh home on Sunday, 21 March.
A full forensic investigation will now be undertaken to establish the cause and nature of her death, although police said there appeared to be no suspicious circumstances.
Following the announcement, the McVey family released a statement, saying: "Over the past ten days we have been living in the hope that Libby would be coming home to us, and we are absolutely devastated that we will never see her again.
"Since she went missing we have had tremendous support from the local community, whose messages of encouragement have kept us going during what has been an extremely difficult time. We would like to thank everyone for their kindness and support.
"Although she is no longer with us, we are left with fantastic memories of Libby, whose fun-loving nature and zest for life was an inspiration to all of us.
"She was a loving wife and mother, and a great friend to all who knew her. She will be sadly missed."
A police spokesman said: "Our thoughts go out to Elizabeth's family at this difficult time. The family have requested that their privacy is respected, so that they can grieve for Elizabeth in peace.
"We are still currently investigating the circumstances surrounding Elizabeth's death, and anyone with information that can assist our enquiries should contact police immediately."
Anyone with any information can contact Lothian and Borders Police on 0131 311 3131, or Crimestoppers in confidence and complete anonymity on 0800 555 111.