The Justice Ministry said the fight broke out Thursday in the psychiatric unit of Korydallos prison, a maximum-security facility in Athens.
One inmate died of his injuries and eight others were hospitalized outside the prison for theirs. A prison officer also was injured.
The ministry said further details would be announced Friday.
Greece has some of the most overcrowded prisons in the European Union. But the number of inmates per cell has improved in recent years due to prisoners having more opportunities for early release, as well as the construction of new prisons and additions to existing ones.