Oleg Sokolov, 63, was drunk and fell into the river as he tried to dispose of body parts, Russian media said.
Police then found the decapitated body of Anastasia Yeshchenko, 24, at his home in the city of St Petersburg.
Prof Sokolov is a Napoleon expert who has received France’s Légion d’Honneur.
“He has admitted his guilt,” his lawyer Alexander Pochuyev told AFP news agency, adding he regretted what he had done and was now co-operating.
Prof Sokolov reportedly told police he had killed his lover during an argument and had then sawn off her head, arms and legs.
He is said to have planned to get rid of the body before publicly taking his own life dressed as Napoleon.
Mr Pochuyev said Prof Sokolov – who has been receiving hospital treatment for hypothermia – may have been under stress.
What do we know about the victim?
Ms Yeshchenko moved to St Petersburg to study from Krasnodar region in southern Russia, and was a postgraduate student at the time of her death.
“She was quiet, sweet and always the ideal student,” an acquaintance told Ria news agency. “Absolutely everyone knew about their relationship.”
She studied French history and co-authored several historical works with Prof Sokolov. The couple enjoyed wearing period costumes, with Prof Sokolov dressing up as Napoleon.
Russian media report that her mother was a police lieutenant colonel and her father a school PE teacher. She had a brother who once played as a goalkeeper for the national junior football team.
The papers here are full of details of this murder, the tragic and gruesome tale of a brilliant and beautiful young student shot and dismembered by her 63-year old former professor-turned-lover.
In these accounts, Oleg Sokolov emerges a historian whose interest in Napoleon bordered on the obsessive. He had a glittering CV as a respected expert on French military history who’d been visiting professor at the Sorbonne. He was also a major figure in the world of historical re-enactment.
One friend told Komsomolskaya Pravda newspaper that he’d organised costume balls and picnics, as well as recreating battles.
But organisations he’s been linked to have now scrubbed his name from their websites.
Anastasia Yeshchenko’s three-year-long affair with the married professor was common knowledge at the university. Friends say she was a top student, highly intelligent, who shared her lover’s passion for Napoleonic history. One person described the professor as “eccentric but not aggressive”; others claimed he considered himself Napoleon reincarnate.
Anastasia’s brother told RBK Media that she called him in tears in the early hours of Friday morning, saying the couple had had a furious row sparked by the professor’s jealousy. She was planning to spend the night in a student hostel.
At 01:49 the two spoke again, and Anastasia told her brother she was fine. It was the last anyone heard from her.
And the suspected killer?
Students described Prof Sokolov as both a talented lecturer who spoke French and did impressions of Napoleon and as a “freak” who called his lover “Josephine” and asked to be addressed as “Sire”, AFP reported.
He was also a member of France’s Institute of Social Science, Economics and Politics (Issep), which on Saturday said it had removed him from his position on its scientific committee.
“We learn with horror… the atrocious crime of which Oleg Sokolov is guilty,” it said in a statement.
“We could never imagine that he could commit such an odious act,” the statement added.
Issep was founded by Marion Maréchal, the niece of far-right National Rally party leader Marine Le Pen and a former lawmaker for the far-right National Front party.