The first victim, also being treated at the University Hospital in the northern city of Lille, was said to be in a "very serious but stable" condition as health officials attempted to play down public concern over two cases that are attracting intense media coverage.
The virus, known as nCoV-EMC, is a cousin of Severe Acute Respiratory Syndrome (SARS), which killed nearly 800 people in the Far East in 2002-03. The new virus is known to have killed at least 18 people, mostly in Saudi Arabia.
The latest French victim shared a hospital room with a 65-year-old man for three days before the latter was diagnosed with the virus, which it is thought he contracted while on holiday in Dubai last month.
Professor Benoit Guery, a specialist in infectious diseases at the hospital, said the transmission between patients should not be taken as an indication that the virus was more contagious than previously thought.
"Thankfully it is not comparable to SARS," he said. "It has been circulating for a little less than a year, and we're at 34 reported cases, whereas with SARS it only took a few months to get to 8,000 cases."