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The health ministry said one Christian man was killed. The ambulance service said he was struck with birdshot.
Mourners inside the church had earlier chanted slogans against Egypt's Islamist President, Mohammed Morsi.
Witnesses told local TV stations that the violence started when a mob attacked mourners as they exited the cathedral, pelting them with stones and petrol bombs. There was initially little police presence.
The Christians responded by throwing stones back, the witnesses said, until police arrived and attempted to quell the unrest, firing tear gas into the cathedral compound.
It also said President Morsi had denounced the violence in a phone call to the head of the Pope Tawadros II, the head of the Coptic Christian church.
"Any attack against the cathedral is like an attack against me personally," he was reported as saying.
Egypt's minority of Coptic Christians, who make up about 10% of the population, have accused the government of failing to protect them, following the fall of Hosni Mubarak in 2011.
Clashes between Muslims and Coptic Christians have been seen numerous times since then, but this weekend's violence was the worst seen in several months.
Meanwhile, Egypt's top judicial body has urged the chief prosecutor appointed by Mr Morsi to step down.
Talaat Abdullah, who was named to the post by President Morsi in December, has provoked anger by demanding the arrest of several high-profile political activists.
In a statement on Sunday, Egypt's Supreme Judiciary Council urged Mr Abdullah to return to his previous job as a judge.
Last week a court annulled the presidential decree that appointed him, but Mr Abdullah continued to carry out his duties, including issuing arrest warrants for activists accused of insulting President Morsi and Islam.
Discontent with the government has also spread to the economy. Unemployment is continuing to rise and industrial action has become commonplace.