Police say he went running along the East River in downtown Manhattan with his academy class on Monday. After completing the run he collapsed and died a short time later.
Bloomberg says Abrams had been a correction officer for four years until he got the call to join the NYPD.
The mayor and Police Commissioner Ray Kelly met with the officer's family at Beth Israel Medical Center on Monday night.
Bloomberg said they offered condolences and thanks on behalf of the city for the officer's service.
The medical examiner will determine the cause of death.
Mayor Michael Bloomberg released a statement about the tragic incident:
"Earlier today, Police Officer Sherman Abrams, who entered the Police Academy this July, collapsed after completing a run with his academy class and passed away shortly afterwards. He was only 28 years old. Sherman made the decision to dedicate his life to protecting New Yorkers, serving as a correction officer for four years until he got the call to join the NYPD. Commissioner Kelly and I met with his mother tonight at Beth Israel Hospital, where we offered our deepest condolences and the thanks of an entire city grateful for his willingness to serve and help keep us all safe. My thoughts and prayers will be with all of Sherman's family and friends."
Police Commissioner Ray Kelly also released a statement:
"Sherman's dedication as a New York City police officer began when he took the oath of office and started his training in the Police Academy. He made a commitment to public service first as a correction officer and then as a police officer with a seriousness few are prepared to make so early in their careers. On behalf of all the members of the New York City Police Department, I extend my deepest condolences to Sherman's family and friends at this difficult time."
(The Associated Press contributed to this report)