The Moscow Times reported Monday that the despite the bill's changes and Duma's approval of 310-120, some said the bill was still too restrictive and so they would work to defeat it.
"We still hope to stop the bill," said Sergei Reshulsky of the Communist Party, adding that Communists will organize demonstrations against the bill.
The Duma is Russia's federal legislature.
The amended bill bans demonstrations from presidential residences, courts, jails, hazardous materials facilities, pipelines, railroads and border areas.
It also lowers the age requirement of rally organizers from 18 to 16 and allows demonstrations at government offices, schools, hospitals, concert halls and embassies.
While the bill eases procedures to hold a rally it leaves the door open for officials to block protests, the Times said.