“He was found with a letter from a psychiatric clinic and we are leaning towards treating it as a mental health case,” Tarabeux told AFP.
The vehicle first drove at speed towards a bus stop in the city's northern 13th district at around 9:00 am (0700 GMT), leaving one person seriously hurt.
The driver then continued onto the eastern 11th district, where he drove at another bus stop, causing one fatality.
The incident comes as police across Europe search for the driver of a van that mowed down pedestrians in Barcelona last week, killing 13.
Several European cities, including London, Berlin and Stockholm, have been targeted in a wave of attacks by Islamist radicals using vehicles to run down people.
Terrorism experts have warned that the intense media coverage of the attacks could spur copycat attacks by people with mental health problems that include a propensity for violence.
Julien Ravier, mayor of the 11th and 12th districts, told the BFMTV news channel that the victim in the Marseille incident was a woman in her 40s who was waiting alone for a bus.
A police source, who asked not to be identified by name, said the driver was in his mid-30s and was not from Marseille.
The local La Provence newspaper reported that he was known to police for minor offences.
BFMTV reported that a bystander noted the registration number of the van, which the police used to trace the vehicle to the city's Old Port district, where the suspect was arrested.
Police sealed off the port area and urged residents in a tweet to avoid the neighbourhood, which is popular with tourists.
Police forensic teams also combed the area around the bus stops for evidence.
The bloodiest vehicle attack in Europe took place in the French city of Nice in July 2016, when a radicalised Tunisian drove a truck through crowds celebrating France's national holiday, killing 86 people.
The Islamic State group has claimed responsibility for several of the attacks.
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