The New Jersey-born actor, married to CSI star Marg Helgenberger, was elected president on Friday at the actors' union, which represents 100,000 actors across the United States.
Little House on the Prairie star Gilbert decided to step down earlier this year after two terms, citing rifts within the union.
Rosenberg says, "I am honored that the members of this great union have placed their confidence in me.
"I ran a campaign that offered a simple and straightforward promise - I will fight like hell to get actors their fair share."
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