Emergency services rushed to Defender Road, in Southampton, just after 5am on Sunday. The baby was taken to hospital where he later died.
A 16-year-old boy and an 18-year-old woman have been arrested in connection with the death and are in custody for questioning.
A Hampshire Police spokesman confirmed that the pair had been arrested on suspicion of murder.
Neighbours reported that the baby died at a unit for young mothers and their children.
Local resident Ian Fussel said: "My friend who lives in a flat heard the baby crying at night. It's not nice at all.
"It's terrible, horrible, disgusting. I've got a three-year-old and a two-month-old, it's wrong, disgusting."
He added: "It's a mother and baby unit, teenage kids live there and there used to be staff there at night but they stopped that."
A police statement said: "Officers investigating the death of a six-week-old boy in Southampton have arrested two people.
"Police were called at 5.05am on Sunday to an address in Defender Road, Southampton, along with South Central Ambulance Service following reports that a child was in need of medical attention.
"The baby was taken to Southampton General Hospital where he was pronounced dead a short time later. Inquiries into the circumstances are ongoing.
"An 18-year-old woman and 16-year-old boy, both from Southampton, have been arrested in connection with the incident.
"They remain in custody at this time and are assisting police with their inquiries."