Two drivers have died with Covid-19 in Milton Keynes, which is experiencing a spread of coronavirus among refuse workers.
The council said "nearly 40%" of staff had been unavailable to work. Some collections have been scrapped.
Serco said the two workers were from different teams and they would be "sorely missed".
Food and green waste collections have been suspended, but black bins and recycling will still be collected.
Mark Sturgeon, Serco's senior contract manager, did not provide further details about the workers who died but offered his "deepest condolences" to their friends and families.
"Sadly we have recently lost two of our colleagues on our Milton Keynes team to Covid-19," he said.
He said the safety of employees was its "first priority" and that "all appropriate measures in line with government guidelines" were being taken.
"In addition, our workplaces and activities are risk assessed and we carry our regular reviews to ensure the highest standards are maintained," he added.
In a message on its website, Milton Keynes Council said: "Our waste collection partner Serco has seen an increase of Covid-19 positive tests and staff requiring to isolate or shield.
"The situation is one that changes rapidly. In recent days nearly 40% of the waste collections teams have been unavailable."
It said the suspension would be "until further notice".
Labour councillor Emily Darlington, the cabinet member for public realm, said the situation was being "taken very seriously".
"I need reassurances that proper things are being put in place to make sure that staff on site, and while they are out working, are safe," she said.
"We are asking very important public health questions and will continue to make sure it is always in-line with public guidance."
Conservative environment and waste spokesman Dan Gilbert said the council had been warned that the waste collection service would come under pressure in the winter.
"They ignored those warnings and now the service is facing significant shortages due to the virus' impact, and there are wide warnings of missed collections," he said.