Under a plea deal, he avoids a possible three-year prison term but must pay $1,000 in fines and can never again run for public office in California.
Velez was elected to the Hawthorne council in 2003 and did not seek re-election last November.
He was accused of voting to approve a project for a developer whose wife owned a home he was renting.
His attorney said the councilman had "no criminal intent."
