It is possible more deaths will be discovered in the coming days as emergency services continue to search the scene, Bedfordshire Police chief superintendent John Murphy warned.
Two people were seriously injured, while a firefighter suffered smoke inhalation after the fire at the block of flats in Redwood Grove.
East of England Ambulance Service (EEAS) also said there had been three hospital admissions, one of whom was airlifted to Addenbrooke’s Hospital with leg injuries.
“Very sadly we know that at least one person has passed away in the explosion today,” Mr Murphy said.
“There is a possibility that there will be more fatalities discovered in the coming days as we continue to search through the scene and a number of people, as my colleague from the fire services has identified, were injured as well today.”
Bedfordshire Fire and Rescue Service earlier described the blaze as a “major incident”, while a local resident said they saw one person leap from a second-floor window to escape the burning building.
Adam Fisher, a 37-year-old sales worker whose office is nearby, said he heard a “huge explosion” before witnessing “an inferno” at the site and people screaming.
“A huge explosion heard - everyone went to our work car park to see what was happening,” he told the PA news agency.
“Our building is adjacent to the apartment complex - immediately an inferno (was) visible, huge smoke cloud billowing, people screaming and people frantically calling for emergency services.”
Another resident, who spoke on condition of anonymity, said they saw a person jumping from a second-floor window to escape the blaze.
“I heard the explosion from my house round the corner,” she told PA.
“I walked to town and saw the flames and also witnessed someone jump from the second-floor window while flames were blaring out.”
The fire service said in a statement: “Bedfordshire Fire and Rescue Service are currently in attendance at a major gas explosion on Redwood Grove, Bedford.
“There are currently six appliances, two water carriers, an incident command unit at the scene.
“Please avoid the area and keep windows and doors closed.”
Bedfordshire Police said in a statement: “(The fire) has engulfed the whole building and caused a large portion of the roof to collapse.”
“An emergency assistance centre has been set up by Bedford Borough Council, at the John Bunyan Centre in Bedford, for people who have been evacuated and need support.
“An adjacent block of flats has also been evacuated and we are working as quickly as we can to get people back into their homes as quickly as possible.”
An EEAS spokesman said: “We were called at 9.33am today to a significant fire at a block of flats in Redwood Grove, Bedford.
“Three ambulances, the hazardous area response team, a tactical commander, three operations managers and the Magpas Air Ambulance were sent to the scene.
“One person was airlifted to Addenbrooke’s Hospital for treatment for leg injuries, while two people, one of whom was a firefighter, were conveyed to Bedford Hospital South Wing with smoke inhalation.”
Nearby Shackleton Primary School was evacuated as a result of the fire, with headteacher Andy George saying in a statement: “Following the fire in the Redwood Grove area of Bedford earlier today, pupils at Shackleton Primary have been sent home due to the school’s proximity to the incident.
“All children are safe and the school is working with the emergency services.
“The school expects to re-open tomorrow.”
With additional reporting from the Press Association