Story highlights FIRST ON CNN : A man on a boat in Havana tells CNN he is Josh Hakken

: A man on a boat in Havana tells CNN he is Josh Hakken Cuba says it will turn the Hakkens back over to the United States

Authorities believe Hakken abducted his two small sons last week

The boys were taken from their parents by child welfare officials in Louisiana last year

The smallest boat moored at the Hemingway Marina in Havana may hold the most intrigue.

Hunkered down inside a blue, 25-foot sailboat named Salty are Josh Hakken and his wife, Sharyn, and their two boys, 2-year-old Chase and 4-year-old Cole.

Theirs is no ordinary visit to the historic port, where they have eluded capture but where CNN found them Tuesday. The Hakkens have been on the lam after they allegedly snatched the two boys from their grandmother's home in Florida.

The couple lost custody of their children last year. There is an international manhunt for this family, and here they are, blending in among the other boats.

But their stop appears likely to be temporary, as Cuban officials announced Tuesday afternoon that they plan to turn the family over to U.S. authorities.

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Josh Hakken glared through his sunglasses at the CNN reporter who found him and said nothing beyond confirming his identity.

The two boys are OK, said a woman matching the description of his wife, and she left it at that.

Josh and Sharyn Hakken are wanted by U.S. authorities and were not in custody in Cuba. But that doesn't mean they aren't being watched.

Cuban security officials wearing sidearms appeared as the CNN video crew was filming and ordered them to stop. At the security guards' request, CNN stepped away from the boat, which looked just like the photo that Florida law enforcement officials had circulated, except more battered.

A statement from the Cuban foreign ministry said the boat put in to the marina, located a few miles west of Havana, in bad weather on Sunday.

"From the first moment, diplomatic notes were exchanged and a permanent and professional communication has been maintained between MINREX" -- Cuba's foreign ministry -- "and the U.S. Interests Section in Havana, with the goal of guaranteeing the integrity and well-being of the minors," the statement said.

U.S. government officials have told the sheriff's office in Hillsborough County, Florida -- which includes the city of Tampa from which the boys went missing -- that "they are receiving exceptional cooperation from the Cuban government," the sheriff's office said Tuesday afternoon.

Earlier in the day, U.S. State Department spokesman Patrick Ventrell said the State Department is aware of the case, though it could not provide additional information because of privacy concerns.

"But what I do want to say, more broadly speaking, is that one of the department's highest priorities is the welfare of U.S. citizens overseas, and this is particularly true for children who are our most vulnerable citizens," Ventrell said.

Some worried that recovering the children will be difficult.

"Unfortunately, these parents and these poor children, these innocent ones, will now be in a country where there are no laws, there is no redress, and that has been a refuge for fugitives and wanted criminals for many years," Rep. Ileana Ros-Lehtinen, R- Florida, told CNN.

The boys had been removed from the care of the Florida couple last year, and on April 2, the couple's parental rights were terminated in Louisiana, investigators say.

The Hillsborough County sheriff's office described Josh Hakken as an anti-government protester, and he was believed to be armed, according to the Hillsborough County sheriff's office.

There is no extradition treaty between Cuba and the United States, although there have been recent cases in which Americans sought for crimes in the United States and discovered in Havana have been sent back by the Cubans.

The FBI currently estimates there are around 70 fugitives from U.S. justice in Cuba, which is one of the reasons Cuba remains on the U.S. list of countries that support state terrorism. Most of the American fugitives in Cuba have been there for decades and have ties to revolutionary movements or radical groups.

Last week, the police department in Slidell, Louisiana, issued its own statement offering background on the Hakkens and why the boys were taken from the parents last year.

In June of 2012, Slidell police responded to a disturbance report at a hotel where Josh and Sharyn Hakken were staying with their sons, the police statement said.

"When police arrived, both Mr. and Mrs. Hakken were acting in a bizarre manner that alarmed officers. They were talking about 'completing their ultimate journey' and were traveling across the country to 'take a journey to the Armageddon'," the Slidell police statement said, adding, "Let it be noted that both of their children were present in the hotel room at the time."

Because of the parents' behavior and "the fact that narcotics and weapons were located inside of the hotel room," the children were taken by child welfare officers, and Joshua Hakken was arrested on drug charges, the statement said.

"Approximately two weeks later, Slidell Police were notified that Mr. Hakken had shown up to the foster family home ... with a firearm demanding the return of his children," the Slidell police statement continued. "The foster parents called 911, and Mr. Hakken fled without his children. We have heard nothing until (Wednesday)."

At some point over the past few months, the children were sent to live with their grandmother, Patricia Hauser, the mother of Sharyn Hakken.

Sheriff's investigators say Josh Hakken entered Hauser's home at 6:30 a.m. last Wednesday. She told police that he tied her up and fled with the children in her silver 2009 Toyota Camry. That vehicle was found later that day just a couple of blocks away from the home.

Those investigators told CNN they believe Hakken joined up with his wife, who was waiting in their pickup truck, and the family drove to a parking garage. A short time later, investigators said, Hakken is believed to have taken a sailboat out of a private slip in nearby Madeira Beach.

Surveillance images showed the boat sailing into the Gulf of Mexico about three and a half hours after the boys disappeared from their grandmother's home, investigators said, adding that the photos showed adults and children on board.