The mother, 30, died at the weekend, reported the Aftonbladet newspaper on Wednesday.
She gave birth last Sunday at the private Sophia maternity clinic and has been in a coma since then. While the eight hospital staff members in the room were preoccupied with the baby, the woman's husband noticed something was amiss, Sveriges Television (SVT) reported.
The clinic lacked facilities for intensive care, so the woman was driven to Karolinska Hospital in Solna suffering a reported cardiac arrest.
The hospital released a statement on Wednesday defending its staff and intensive care services. A spokesman said that an investigation had been launched, and that more answers would be available by Friday.
The Sophia Maternity Hospital (BB Sophia) opened in April earlier this year, and concerns were expressed at the time over the clinic's lack of resources for intensive care.
It is understood that the hospital has some intensive care facilities, including two intensive care beds, two operation rooms, and anaesthesia with full on-call staff for intensive care.
Reports said the woman's child was healthy.