The constable's sample was sent for COVID-19 test and the reports came as positive on Wednesday, they said.
This is the first casualty in Delhi Police with respect to the COVID-19 pandemic.
More than 26 police personnel of the force have been tested positive for the virus so far. Some of them have resumed their duties post recovery.
Taking to Twitter, Delhi Police Commissioner S N Shrivastava expressed his condolence to the family of the deceased and assured of all assistance.
"The sudden demise of late Constable Amit Kumar from PS Bharat Nagar has saddened the police fraternity. We stand by his family in this hour of grief and pray to the Almighty to provide strength to bear this loss. All assistance to his family will be provided," he tweeted.
The constable, a resident of Sonipat in Haryana, was posted at the Bharat Nagar Police Station in northwest Delhi, police said.
He had fallen sick on Tuesday and was examined at the Deep Chand Bandhi Hospital where he was given medicines, they said.
On Tuesday evening, when the constable reported that he was feeling uncomfortable, he was immediately taken to the Ram Manohar Lohia Hospital where he was declared brought dead, a senior police official said.
The police personnel who came in contact with the constable have been home quarantined, he said, adding the post mortem examination is yet to be conducted.
The constable is survived by his wife and 3-year-old son, the official added.