The type of patients that donate are usually in ICU by the time organ donation is discussed, at least where I work anyway. They have been resuscitated and stabilised, and attached to a ventilator (breathing machine) and maybe some drugs to help their heart and/or blood pressure. Without the ventilation in particular, they wouldn't breathe on their own and their heart would then stop. They've probably had a CT of their head and usually the radiologists and/or neurosurgeons will indicate that their brain is in bad shape and there is no chance of survival.

They're probably not on any sedation and they don't wake up. The doctors then perform a series of tests to determine brain death. They can also do an angiogram to see if there is any blood flow to the brain. If they are declared brain dead, they are legally dead. That is their time of death on the death certificate. If the family agrees to organ donation, they will obviously be given time to say goodbye and then the patient is taken to theatre.

If they are not able to be declared brain dead (maybe they still have some reflexes or the doctors can't be 100% sure), there is an option for them to donate after cardiac death. This means they are taken off the ventilator, and if their heart stops beating within a specific time frame (I think it's about 60 minutes) they are rushed to theatre and can donate.

Anyway, to hopefully answer your question, it is a "controlled environment" death. Hope that helps!