Earlier this year, a federal judge shut down the free music-download site Mp3skull.com and awarded $22 million to the record companies that had sued it for copyright infringement. But Mp3skull.onl, which has surfaced in its place, is touting a service even more worrisome to the music industry: stream ripping.

That practice, which involves turning a song or music video played on a streaming service into a permanent download, is growing fast among young music fans, even as other forms of music piracy wane. The site’s community...