The Atlantic has a new message for online readers using an ad blocker: disable it, pay up or leave.

The politics and culture magazine is rolling out a new advertising-free digital subscription option for $3.99 a month or $39.99 a year. Those who try to read the Atlantic with ad-blocking software enabled will be prompted to either “whitelist” the site, meaning that ads can filter through, or sign up for the ad-free model.

“Digital...