Thierry Frezier, 60, was freed after collaboration between Sudanese, Chadian and French intelligence services, a member of the National Intelligence and Security Service (NISS) told reporters in Khartoum.
Frezier was taken to Sudan's capital after being freed on Saturday.
French President Francois Hollande's office put out a statement saying he felt "great pleasure" at the release.
Sudanese security agents launched a search for Frezier in late March after a Chadian minister said he was being held there after being abducted near Goz Beida in southeastern Chad on March 23.
His kidnappers took him to Sudan's war-torn region of Darfur and it was from there that he was liberated, a foreign ministry official in Khartoum said.
"I thank Sudan's government and civil and military authorities for their efforts to free me," Frezier said in a brief statement to reporters at Khartoum airport on Sunday.
"The kidnappers treated me well during my time in captivity."
NISS officer Mohamed Hamid said Frezier would be handed over to the French embassy in Khartoum.
"An operation to free him was launched in coordination with French and Chadian intelligence services," Hamid said.
Sudanese and French officials in Khartoum said no ransom had been paid.
"The outlaws inside Chad had kidnapped him for ransom, but no ransom has been paid," Sudanese foreign ministry official Khalid Al-Kalas said on Sunday.
"NISS was monitoring his situation and yesterday they liberated him and also captured his kidnappers."
France's charge d'affaires in Khartoum, Christian Bec, also told reporters that no money had been handed over in exchange for Frezier's freedom.
"I thank the Sudanese government for liberating the hostage without paying any ransom," Bec said.
Several French and other Western nationals have been kidnapped by jihadist groups in west and central Africa in recent years.
The last such case in Chad -- a former French colony -- was in 2009, when a Frenchman working for the International Committee of the Red Cross was abducted by a shadowy armed group called the Freedom Eagles of Africa, based in Sudan's Darfur province.
He was freed after 89 days.
Chad is one of France's key African allies in the counter-terror fight, with its capital N'Djamena serving as headquarters for France's Operation Barkhane anti-jihadist force.
Set up in 2014, the force, which counts 4,000 troops, works in five Sahel countries -- Chad, Mali, Niger, Mauritania and Burkina Faso -- to flush out Al-Qaeda-linked extremists.
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