Do you tell the prosecutor?

You are a homicide investigator and are interrogating someone you believe picked up a 9-year-old in a shopping mall, and then molested and murdered the girl. He is a registered sex offender, was in the area, and although he doesn’t have any violence in his record, you believe he must have done it because there is no other suspect who had the means, opportunity, and motive. You have some circumstantial evidence (he was seen in a video following the child) but very little good physical evidence. You really need a confession to make the case. You want to send this guy away for a long time. After several hours of getting nowhere, you have a colleague come in with a file folder and pretend that the medical examiner had obtained fingerprints on the body that matched the suspects. You tell him that he lost his chance to confess to a lesser crime because now he is facing the death penalty. He says that he will confess to whatever you want him to if the death penalty is taken off the table. Do you tell him what you did? Do you tell the prosecutor?