It happened at Harbor House Apartments just before 10:30 a.m.
At the scene, police say they found a 35-year-old man suffering from a gunshot wound.
The victim, identified by police as Devontre Taylor of Wilmington, was taken to a hospital and later pronounced dead.
According to police, authorities later found the suspect after he was seen entering a building in the apartment complex.
The suspect, identified as 27-year-old Michael Toombs of Claymont, was detained at the building and transferred to police headquarters.
Toombs has since been charged with first-degree murder, first-degree robbery, and other related offenses.
Investigators have not yet said what led to the deadly shooting.
Anyone with information on the incident is asked to contact the police at 302-395-2787.