The BBC report that it is not officially condoned but is still practised in several rural areas. The act is still practised by the Sabiny, some Karamojong sub-groups and the Pokot in eastern Uganda and the Nubi people of West Nile.
Genital mutilation is seen in some countries as a way to ensure virginity and to make a woman suitable for marriage.
MP Alice Alaso said the move was "a very significant achievement".
"It's a very bad practice. It's cruel, it traumatises people, it's led children to drop out of school, it's a health hazard," she told the BBC's Focus on Africa programme.
"This is a warning signal - whoever dares practice female genital mutilation will be subject to the law."
Another MP, Lulume Bayiga, said the law would liberate both men and women - who often face being ostracised for shunning the custom.
"Women will start for the first time to enjoy sex and it's going to do away with various diseases," he said.
According to the UN, about three million girls each year in Africa are at risk of genital mutilation, with more than 91 million girls and women living with the consequences of the procedure.
These include bleeding, shock, infections and a higher rate of death for new-born babies.
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