The World Health Organisation has given warning that a new strain of the coronavirus could be passed on between people in close contact.
The warning on Sunday came after authorities confirmed that a man who had shared a hospital ward with France's first victim had been infected. The second patient's condition has deteriorated, requiring treatment in intensive care, Health Minister Marisol Touraine said at a news conference.
The minister added that the first patient was "in a stable but worrisome situation".
The novel coronavirus has killed 18 people since being identified last year, out of more than 30 confirmed cases reported to the World Health Organisation since September 2012.
The first France patient had just returned from vacationing in the United Arab Emirates when he fell ill.
He shared a room with the second patient for a few days in late April at a hospital in Valenciennes. Now both are being treated at the Lille Hospital in northern France.
The coronavirus, known as nCoV-EMC, is a cousin of Severe Acute Respiratory Syndrome (SARS), which killed nearly 800 people in the East Asia in 2002-03.
Most of those infected since the virus was identified had travelled to Qatar, Saudi Arabia, Jordan or Pakistan. There also have been cases in Britain and Germany.
Al Jazeera's Jamal Elshayyal said that the biggest cluster has been found in Saudi Arabia. He said for now the coronavirus is nowhere near as widespread as SARS was 10 years ago.
'Need for awareness'
WHO Assistant Director-General Keiji Fukuda told reporters in Saudi Arabia, the site of the largest cluster of infections,
there was no evidence so far the virus was able to sustain "generalised transmission in communities" - a scenario that would raise the spectre of a pandemic.
But he added: "Of most concern ... is the fact that the different clusters seen in multiple countries ... increasingly support the hypothesis that when there is close contact, this novel coronavirus can transmit from person to person.
"There is a need for countries to ... increase levels of awareness," he said.
Benoit Guery, head of the Lille hospital's infectious diseases unit, 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."
Saudi Deputy Health Minister for Public Health Ziad Memish told reporters that, of 15 confirmed cases in the most recent outbreak, in al-Ahsa district of Eastern Province, nine had died, two more than previously reported.
Saudi Arabia's Health Ministry said in a statement the country had had 24 confirmed cases since last summer, of whom 15 had died. Fukuda said he was not sure if the two newly reported Saudi deaths were included in the numbers confirmed by the WHO.
Fukuda, part of a WHO team visiting Saudi Arabia to investigate the spread of the disease, said although no specific vaccine or medication was yet available for novel coronavirus, patients were responding to treatment.
"The care that is taken in the hospitals, in terms of using respirators well, in terms of treating pneumonia, in terms of treating complications, in terms of providing support, these steps can get patients through this very severe illness," he said.
Fukuda said that as far as he knew all cases in the latest outbreak in al-Ahsa district were directly or indirectly linked to one hospital.