According to police, officers responded to the parking lot of a Papa John's in the 2100 block of Marsh Road for a report of a shooting around 6:45 p.m. Saturday.
Upon arrival, police found a 35-year-old woman suffering from a gunshot wound to the face.
She was taken to the hospital with serious injuries.
Police say an investigation determined that the victim was shot during a verbal altercation with the suspect, who was identified as 39-year-old Paul Waller of Wilmington.
Waller was taken into custody Sunday in Pennsylvania.
He will be extradited back to Delaware, where he will face charges.