Alexandre, an Aristide appointee, is president of Haiti's Supreme Court and is designated by Haiti's constitution to temporarily assume the presidency if the office becomes vacant.
Aristide resigned in the face of an armed revolt and fled the country to neighbouring Dominican Republic. He is now on his way to Morocco, the Haitian consul in the Dominican Republic said.
In a ceremony at Prime Minister Yvon Neptune's residence, Alexandre said he "accepts responsibility with sincerity."
"The president has just given his resignation and in accordance with what is proscribed by the constitution the interim must be assumed by the president of the court," Alexandre said. "Haiti, I exhort you to remain calm," he said. "No one should take justice into their own hands."
Neptune read a statement by Aristide in which he said he resigned to avoid bloodshed.
As news of Aristide's departure spread, reaction in the streets of the capital, Port-au-Prince, was swift and angry.
Armed bands of "chimeres," the most militant of Aristide's supporters from the slums roamed the city in pickup trucks, armed with shotguns and other weapons.
The Aristide militants had been set to give the rebels a tough fight if they did move in on the sprawling city, prompting worries the city could be plunged into a bloodbath.
Aristide's supporters have gone
on rampage in Port-au-Prince
Looters hit a police station in Petionville, an upscale suburb in the hills above the capital, carting away police hats, t-shirts, helmets and other parts of police uniforms.
Two men carried away a new refrigerator, still wrapped in plastic. One balanced a cooler on his head. Others took televisions, VCRs, filing cabinets, even the doors from the police station.
On a major thoroughfare, a barricade constructed of burning tires sent a wall of thick, black smoke along the street.
Aristide supporters drove up and added tires to fuel the fire. Others gathered on street corners, puzzled by the events.
Aristide's departure came under pressure from the United States, France and other nations for him to resign and end the revolt, which has killed nearly 70 people.
The US, which invaded the country in 1994 to restore Aristide to power, has been criticized for acting too slowly in the current crisis. It failed last week to negotiate a compromise between the politicians in Port-au-Prince meant to avert a collapse of the government.
"The kingdom of Morocco has no intention of responding favourably to an eventual demand granting political asylum to President Aristide in Morocco"
Moroccan Foreign Ministry statement
But by the end of the week Washington had made clear it had lost faith in Aristide and his departure was a matter of time.
The Pentagon has warned that 2000 Marines were on alert and could be in Haiti in two days.
Meanwhile, Morocco has said it would not grant the Haitian president asylum.
"The kingdom of Morocco has no intention of responding favourably to an eventual demand granting political asylum to President Aristide in Morocco," said the official MAP news agency, quoting a Foreign Ministry statement.