His father Jeff confirmed the news on Sunday in a Facebook post showing the infant.
"Meet the newest member of our family, Remington Monroe Simms! I'm already wrapped around that tiny little finger," the grateful grandpa wrote.
Speaking with Dr. Drew last month, Corey said his intention to get custody was so his children could be raised "in a stable home" amid Messer's issues, which landed her in rehab for a month earlier this year.
"I don't think that's too much to ask … it's hard for me to get the truth out of Leah about a lot of stuff," he said. "I've been in my house going on four years. I'd like to show them what it's like to be in a place, growing up, stable."