It said more than 125 million girls and women alive today had undergone a procedure now opposed by the majority in countries where it was practised.
Ritual cutting of girls' genitals is practised by some African, Middle Eastern and Asian communities in the belief it protects a woman's virginity.
Unicef wants action to end FGM.
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The UN Children Fund survey, described as the most comprehensive to date on the issue, found that support for FGM was declining amongst both men and women.
FGM "is a violation of a girl's rights to health, well-being and self-determination," said Unicef deputy executive director Geeta Rao Gupta.
"What is clear from this report is that legislation alone is not enough."
'Speak out loudly'
The report, 'Female Genital Mutilation/Cutting: A statistical overview and exploration of the dynamics of change', was released in Washington DC.
The study, which pulled together 20 years of data from the 29 countries in Africa and the Middle East where FGM is still practised, found girls were less likely to be cut than they were some 30 years ago.
They were three times less likely than their mothers to have been cut in Kenya and Tanzania, and rates had dropped by almost half in Benin, the Central African Republic, Iraq, Liberia and Nigeria.
But FGM remains almost universal in Somalia, Guinea, Djibouti and Egypt and there was little discernible decline in Chad, Gambia, Mali, Senegal, Sudan or Yemen, the study found.
However, it did find that most girls and women, and a significant number of boys and men, opposed the practice. In Chad, Guinea and Sierra Leone more men than women wanted to see an end to the practice.
"The challenge now is to let girls and women, boys and men speak out loudly and clearly and announce they want this harmful practice abandoned," said Ms Rao Gupta.
The report recommends opening up the practice to greater public scrutiny so that entrenched social attitudes to it can be challenged.
In some communities FGM, also known as female circumcision, is seen as a traditional ritual used culturally to ensure virginity and to make a woman marriageable.
It typically involves procedures that alter or injure female genital organs and is often carried out by traditional circumcisers, who play other central roles in communities.
The dangers of FGM include severe bleeding, problems urinating, infections, infertility and increased risk of newborn deaths in childbirth.