The woman was shot late Thursday in a University of Illinois at Chicago hospital garage and later died, authorities said. Her name hadn't been released pending notification of her family.
University police characterized the killing as an "apparent domestic-related shooting" in a campus alert posted early Friday on the school's website and said officers were seeking a man who was a housekeeper at the hospital.
Officers pulled the 47-year-old suspect over for a routine traffic stop at 6:30 a.m. Friday, and the officer recognized the man from a description in the alert, university police spokesman Mark Rosati said. The suspect fled the car and was arrested a few blocks away, he said.
A weapon was recovered but officers didn't immediately know if it was used in the shooting.
Rosati said the suspect had worked at the hospital for 15 years, but he declined to discuss his relationship to the woman who was killed.
The hospital was locked down for hours while police searched for the gunman but resumed normal operations around 5 a.m. Friday. Rosati said authorities don't believe the gunman was ever inside the hospital after the shooting.
The garage is near the hospital but not attached to it, Rosati said. He wasn't sure whether the hospital has metal detectors but said the university has security cameras posted indoors and outdoors across its 250-acre campus.