By Alex Weprin Senior Editor Gayle King is staying put at CBS. The longtime anchor on CBS Mornings has inked a new deal with the network to remain on the a.m. show, while also expanding her purview to new projects with the network. The host is one of the faces of CBS News, which is undergoing a restructuring process under the leadership of editor-in-chief Bari Weiss and president Tom Cibrowski. “There is only one Gayle King. We’re so proud that she’ll continue to call CBS home,” Weiss said in a statement. “We’re thrilled to have her on in the morning — and equally excited to work with her on new, enterprising projects that bring her talents to new audiences.” King’s prior contract had been set to end in May, leading to speculation about her future with the network under the new leadership. In fact, King had to deny a report last year that she was going to leave the program. “Rumors of my demise were inaccurate and greatly exaggerated,” King said in a statement.…