Updated at 4:35 p.m. ET

A strong earthquake that hit Mexico City and other central areas has killed at least 273 people, officials say. Search teams are working feverishly to find any survivors who were trapped.

However, Navy Assistant Secretary Angel Enrique Sarmiento said there is no missing child at the collapsed Mexico City school that has been the focus of rescue efforts.

"We've done an accounting with school officials, and we are certain that all the children either died, unfortunately, are in hospitals or are safe at their homes," said Sarmiento as quoted by The Associated Press. He said 19 children and six adults were killed in the school collapse and 11 children were rescued.

Sarmiento said traces of blood and other signs suggest that an adult may still be alive in the rubble.

The stunning announcement follows multiple reports that a girl had been able to communicate with emergency crews and had wriggled her fingers for them in the debris of the Enrique Rebsamen School, south of the capital. The building collapsed during Tuesday's 7.1 magnitude quake.

After visiting the site, NPR's Carrie Kahn reports:

"It is a heartbreaking scene. Hundreds of volunteers and rescue personnel have flooded to this neighborhood around the school ... all are emotionally drained, tired, but just holding on to hope they can reach some of the children alive ... under all that rubble."One wing of the school, three stories just pancaked in the powerful quake. One right on top of the other, making the rescue effort and chances of survival very difficult."But the volunteers keep coming ... with hard hats and fluorescent vests. They're removing the rubble with picks, shovels, their hands...whatever they can. And dozens more are taking in donations, feeding the rescuers, just wanting to be there and do something for those children either dead or trapped in the building."

Rescue workers have spent hours trying to free whoever might have survived. In addition to heavy rubble that sits precariously in the debris pile, the effort has been frustrated by heavy rain that fell overnight.

It is not immediately clear how rescue workers assumed they were trying to free a girl reported to be Frida Sofia. That was the name a doctor who was working with the rescue team told the AP. The doctor added that the girl said there were several other children near her who were also alive. That now appears to have been a mistake.

The name Frida Sofia became a top-trending term on Twitter — but there were questions about the name. Media outlets in Mexico reported it, especially after journalist Joaquín López-Dóriga tweeted it. But teachers had said there was no student at the school with that name — and El Universal reports that a rescuer used the name as a way to communicate with the girl.

Sarmiento's announcement appears to answer one lingering mystery throughout the rescue effort: No family members had emerged identifying themselves as relatives looking for the girl.

All the same, El Universal's main headline on Thursday read, "The hope of Rebsamen is called 'Frida.' "

Authorities say they've pulled dozens of survivors from damaged buildings. But Mexico City Mayor Miguel Ángel Mancera says more than 35 buildings collapsed — from offices and apartments to schools.

Mexico City's metro service says it's allowing people with rescue tools — picks, shovels and mallets — to ride on its vehicles. And for the second day, the service is free.

"President Enrique Peña Nieto has declared three days of mourning for victims of the quake," Carrie reports. "Schools in the capital and surrounding affected states remain closed until Monday."

In Mexico City and outlying areas — including the city of Jojutla, in Morelos state, where many houses and buildings were reduced to rubble — soldiers, police, firefighters and volunteers have alternated between working to find survivors and undergoing pauses of total silence, as rescuers call out for anyone who is still alive to respond.

"While many eyes are on the earthquake effects in Mexico City, this town of 20,000 people was crumbling," James Fredrick reports from Jojutla for NPR. "Its old abode buildings were no match for the 7.1 quake."

Jojutla Mayor Alfonso de Jesus Sotelo "says 2,000 buildings are damaged, 300 of those totally collapsed; 16 people have died, including four children," James says.

"Definitely this loss is unsustainable," Sotelo says. "We are out of control, being able to correct or absorb the cost that's involved."

If initial reports and relief efforts seemed to focus on Mexico City, James reports, "A day later that has changed: Hundreds of young volunteer rescue workers line up to clean rubble. Donated bottles of water and canned food pile up all over the town."

In Puebla, where the epicenter of the quake was located in the western part of the state, Gov. Antonio Gali gave a grim account of the losses on Thursday morning.

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