Addis Ababa city spokeswoman Dagmawit Moges said most of the dead were women and children, and more bodies were expected to be found in the coming hours.
It was not immediately clear what caused Saturday night's collapse at the Koshe Garbage Landfill, which buried several makeshift homes and concrete buildings. The landfill has been a dumping ground for the capital's garbage for more than 50 years.
Many people at the landfill had been scavenging items to make a living, but others live there because renting homes, largely built of mud and sticks, is relatively inexpensive.
An AP reporter saw four bodies taken away by ambulances after being pulled from the debris. Elderly women cried, and others stood anxiously waiting for news of loved ones. Six excavators dug through the ruins.
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The resumption of garbage dumping at the site in recent months likely caused the landslide, Assefa said. The dumping had stopped in recent years, but it resumed after farmers in a nearby restive region where a new garbage landfill complex was being built blocked dumping in their area.
Smaller collapses have occurred at the Koshe landfill in the past two years but only two or three people were killed, Assefa said.
Around 500 waste-pickers are believed to work at the landfill every day, sorting through the debris from the capital's estimated 4 million residents. City officials say close to 300,000 tons of waste are collected each year from the capital, most of it dumped at the landfill.
Since 2010, city officials have warned that the landfill was running out of room and was being closed in by nearby housing and schools.
City officials in recent years have been trying to turn the garbage into a source of clean energy with a $120 million investment. The Koshe waste-to-energy facility, which has been under construction since 2013, is expected to generate 50 megawatts of electricity upon completion.
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