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An official report said up to 20,000 had died and many tens of thousands have been injured or are missing.
The main hospital, a big department store, the central mosque, a girls' school and two housing complexes have been destroyed.
In 1954, an earthquake killed 1,657 people in El Asnam, which was then called Orleansville.
Risk of typhoid
The first quake hit just before 1330 local time. It measured 7.3 points on the Richter scale - the biggest ever recorded for that part of Algeria.
Three hours later there was a second quake registering 6.3 points.
Telephone lines between El Asnam and the capital, Algiers, have been cut.
The damage to the local hospital is so bad that casualties are being sent more than 100 miles (160km) to Algiers and the northwestern port of Oran for treatment.
In the town centre where the damage is worst, whole blocks of flats have been reduced to heaps of rubble.
A central four-storey hotel collapsed under the weight of its roof, demolishing all 1,509 rooms beneath and burying many of the occupants.
A lot of buildings collapsed, but their flat roofs made of reinforced concrete remained intact, crashing down on top of the crumpled walls and creating tomb-like spaces beneath.
Within hours of the earthquake, many giant earth-moving machines were on the scene helping remove the rubble that buried so many people.
Most of the bodies have been taken to the grounds of the local hospital for identification.
The rescuers are trying to recover the bodies as quickly as possible because of the risk of a typhoid epidemic.
A mass vaccination programme is already planned for later in the week while demolition teams are regularly disinfected to try to limit the likelihood of the disease occurring.
The town's population has tripled since the last quake in 1954, with the result that although this was a less powerful upheaval, it caused much more concentrated damage.
The town centre was home to 150,000 people and another 50,000 lived in the surrounding villages, where hardly a building has been left standing.
Doctors and medical staff have been working without a break since the quake struck but in the outlying areas many more doctors and medicines are needed.
The final death toll was much less than the 20,000 or so first predicted - around 3,500.
There were a number of aftershocks after the first two quakes which sent more refugees fleeing to the relative safety of the countryside.
Mountain sniffer dogs from the French and Swiss Alps were brought in to help look for survivors beneath the rubble - a number of whom were found.
El Asnam was subsequently rebuilt and renamed Chlef.
In May 2003 another earthquake hit parts of Algiers and Boumerdes. More than 2,250 people were killed and 10,000 injured.
The government announced it would spend £1.8bn in a two-year programme to construct new housing.
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