Thursday’s ruling also requires the government to submit to Judge Morin a list of all the data it is requesting and a justification for why it should be included under the warrant.

Mr. Aghaian said the scope of the Justice Department’s search now extended only to emails passing through DisruptJ20.org and related mailing lists. But he warned that the ruling on Thursday could still have a chilling effect on political speech.

The American Civil Liberties Union said on Twitter that despite DreamHost’s “strong pushback” against the Justice Department, the warrant still threatened First Amendment rights. “We’re watching,” the group wrote.

As for the cyberattack against DreamHost, which began Thursday morning, the company said it had restored all services by late afternoon.

The assault began soon after DreamHost began hosting the website Punished Stormer, which is the newest manifestation of the neo-Nazi website The Daily Stormer. The website has spent the past week bouncing from host to host, with GoDaddy, Google and several others refusing it access after the violence at a white nationalist protest in Charlottesville, Va., this month.

DreamHost also hosts websites for the white nationalist groups Northwest Front and National Vanguard. Years ago, the company hosted The Daily Stormer as well, though DreamHost told The New York Times that it later asked the website to leave after it violated the terms of service.

Early Thursday, a user on the social media network Gab identifying himself as Andrew Anglin, the founder of The Daily Stormer, wrote that he had “stopped registering random names because just going on and off was stupid and pointless.”