By Laura MacInnis

NEW YORK (Reuters) - President Barack Obama and first lady Michelle Obama on Thursday toured the soaring skyscraper being built to replace the twin towers destroyed on September 11, 2001, in a poignant visit to mark a new chapter of recovery from the traumatic attack.

Inscribing one of the steel beams that will be part of the building's framework, Obama wrote: "We remember, we rebuild, we come back stronger!" followed by his signature.

Obama has touted the killing of Osama bin Laden, the al Qaeda leader behind the attacks on the United States, as one of his top national security achievements and refers to it often as he campaigns for re-election.

The president and first lady looked down on the 9/11 memorial from One World Trade Center's 22nd floor, which is still under construction, after walking beneath bare pipes illuminated by light bulbs strung from the unfinished ceiling.

"We couldn't be prouder of you guys. This is what the American spirit is all about," Obama told construction workers after a brief outdoor ceremony in the shadow of the building, which is scheduled to be completed in late 2013.

Obama visited the site last September to commemorate the 10-year anniversary of the 2001 attacks by al Qaeda that killed nearly 3,000 people and destroyed the landmark twin towers.

In April, One World Trade Center surpassed the Empire State Building as New York's tallest structure, with a horizontal beam laid at a height of 1,270 feet.

When completed, it will rise 1,776 feet to the tip of its antenna, taller than the World Trade Center towers. Construction began six years ago on the edifice, which was formerly called the Freedom Tower.

New York Governor Andrew Cuomo has called the building "a symbol of the enduring spirit of the city and state of New York."

(Reporting By Alister Bull; Editing by Eric Walsh)