The video was recorded Monday and posted to social media, where police were able to see it, KTRK reported.
Walmart officials told police they saw Anderson remove the carton and bought it before anyone else could purchase it.
Anderson and his father returned to the store to show the receipt proving that he had paid for it, KTRK reported.
Anderson's father said his son did the stunt for Facebook likes, according to KTRK.
Jefferson County District Attorney Bob Wortham said the store still replaced the ice cream and the charges were filed due to the financial loss the store suffered and the cost to law enforcement for the investigation.
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Anderson faces up to a year in jail and a $4,000 fine, KDFM reported.
Wortham also explained another reason Anderson is in trouble with the law.
"This guy loves publicity, even if it's for the wrong reason. That can lead to bigger and worse things. If we're going to save this guy, we have to do it with this case or he's over the hill," Wortham told KDFM.