BRUSSELS — Serbia’s ambassador to NATO was chatting and joking with colleagues in a multistory parking garage at Brussels Airport when he suddenly strolled to a barrier, climbed over and flung himself to the ground below, a diplomat said.

By the time his shocked colleagues reached him, Branislav Milinkovic was dead.

His motives are a mystery. Three diplomats who knew Milinkovic said he did not appear distraught in the hours leading up to his death Tuesday night. He seemed to be going about his regular business, they said, picking up an arriving delegation of six Serbian officials who were to hold talks with NATO, the alliance that went to war with his country just 13 years ago.

“It was indeed a suicide,” Ine Van Wymersch of the Brussels prosecutor’s office said. She said no further investigation was planned.

The diplomats, who spoke on condition of anonymity because they were not authorized to release details, said they knew of no circumstances — private or professional — that would have prompted him to take his own life. Milinkovic, 52, had mentioned to colleagues at diplomatic functions that he was unhappy about living apart from his wife, a Serbian diplomat based in Vienna, and their 17-year-old son.