Coptic mourners leaving Cairo’s main cathedral are said to have clashed with local residents.
Police fired tear gas to break up the violence.
The head of Egypt’s national ambulance service, Mohammed Sultan, said one person had died of birdshot wounds.
Mourners inside the church had earlier chanted slogans against Egypt’s Islamist President, Mohamed Morsi.
Witnesses told local TV stations that the violence started when a mob attacked mourners as they exited the cathedral, pelting them with rocks and petrol bombs.
The Christians responded by throwing rocks back, the witnesses said, until police arrived and attempted to quell the unrest.
Egypt’s state news agency said the streets around St Mark’s Cathedral had seen “on-and-off” clashes between Christians and “unidentified persons”.
It was reported that a fire had started in a building adjacent to the Cathedral, but the blaze had since been extinguished.
Pope Tawadros II, the head of the Coptic Christian church, appealed for calm and the preservation of national unity.
Speaking on Sunday evening, he said he was in contact with government officials.
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.
Police said five deaths – four Copts and one Muslim – occurred on Saturday in Khosous, about 10 miles north of Cairo, after inflammatory symbols were drawn on an Islamic institute, provoking an argument.
The dispute escalated into a gun battle between Christian and Muslim residents, while Christian-owned shops were also attacked.
Violence there flared again on Sunday, with police reporting more sectarian fighting on the streets and clashes between police and youths.
Meanwhile, Egypt’s top judicial body has urged the chief prosecutor appointed by Mohamed Morsi to step down.
Talaat Abdullah, who was named to the post by President Mohamed 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 Talaat Abdullah to return to his previous job as a judge.
Last week a court annulled the presidential decree that appointed him, but Talaat Abdullah continued to carry out his duties, including issuing arrest warrants for activists accused of insulting President Mohamed Morsi and Islam.
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