Fighting at Koutoukale, 50km (31 miles) north-west of the capital, Niamey, was reported at dawn.
Mr Amadou told the BBC the attackers were from the Mali-based al-Qaeda splinter group Mujao.
Last week, German Chancellor Angela Merkel was in Niger in part to offer security help.
The prison is holding jihadists from both the Nigeria-based Boko Haram group and those linked to al-Qaeda.
But the government spokesman said that no-one had escaped during the attack.
"The attack has been contained. One attacker has been killed. They are now being hunted," he added.
The BBC's Himadou Hamadou in Niamey says a local journalist told him that the attackers had come from the direction of the border with Mali.
Mujao, or the Movement for Unity and Jihad in West Africa, is one of several Islamist militant groups based in Mali.
Niger has in the past been targeted by Boko Haram and al-Qaeda-linked militants.
The country's army is part of the multi-national force fighting Boko Haram in Nigeria and its neighbours.
An attack by Islamist militants on a high-security prison in Niger holding about 100 jihadists has been foiled, government spokesman Marou Amadou says.