The U.S. House of Representatives passed a bill Wednesday that would grant legal marijuana businesses access to banking, a measure that would clear up a longstanding headache for the industry.

The bill, called the SAFE Banking Act, passed 321-103 on the strength of near-unanimous support from Democrats and nearly half of Republicans. Its prospects in the Senate are uncertain, but supporters said the amount of Republican support in the House was a good sign.

"This is a sign the time has come for comprehensive cannabis reform," said Morgan Fox, a spokesman for the National Cannabis Industry Asso...