NEW YORK CITY ? The government agency that owns the World Trade Center site is taking over construction of the Sept. 11 memorial, following recommendations that the move could trim spiraling costs.

But the chairman of the Port Authority of New York and New Jersey also said the agency could back out of the agreement if the price of the project exceeds the latest estimate of $510 million.

The boards of the Port Authority and the World Trade Center Memorial Foundation voted Thursday to have the Port Authority build the memorial instead of the nonprofit memorial foundation.

Officials already have approved a new, less-expensive version of the “Reflecting Absence” design after contractors estimated in May it would cost close to $1 billion.

Preliminary memorial construction began in March and stopped two months later when the cost estimates came out.