Yes a Weasel can easily get through chicken wire so the chicken wire is not sufficient enough to keep the weasels out of the chicken coop.
You can keep weasels out of a chicken coop by using some hardware cloth and because the small holes in the hardware cloth the weasels won't be able to get through to your chickens.
The hardware clot has those tiny enough holes to allow airflow into the chicken coop but it will keep the weasels out.
So you should place some hardware cloth over the windows or any openings of the chicken coop if you have a problem with Weasels.
My grandma had trouble with weasels getting to her chickens and after we put hardware cloth on the chicken coop windows the weasels couldn't get through anymore.