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The piggy-bank approach to marijuana sales also creates problems for state and local agencies charged with collecting taxes. “I don’t know how you regulate or tax and industry where you can’t follow the money,” said Mr. Childears.

Burden of bounty

The Denver City Council approved a resolution last week urging federal authorities to help resolve the issue, while Rep. Ed Perlmutter, Colorado Democrat, has introduced the Marijuana Businesses Access to Banking Act of 2013.

Mr. Elliott said he worries about safety, with thieves or crime syndicates targeting Colorado’s cash-rich but banking-poor pot shops.

“We’ve got members, in some cases they’re beautiful young women, who are forced to carry around a whole lot of cash,” said Mr. Elliott. “I’m having nightmares about it.”

Many store owners have hired security and installed cameras to enhance safety. The Denver Police Department increased its presence around the stores during the first week at the request of business owners.

“That was an incredible thing, because the police community and the marijuana community have had no common ground for decades,” said Mr. Elliott. “And now I have their cellphone numbers, and when they come by, they’re greeted with a handshake.”

The lack of dependable banking aside, Mr. Cullen said, he “couldn’t be happier” about his marijuana business. He has hired five more people since Jan. 1 and is finishing construction on two warehouses, which will require another 20 employees.

He said his two Denver retail stores are averaging one sale per 90 seconds. His biggest challenge is “figuring out how to give everyone a lunch break.”

“Everything from the customers to the staff who’ve stepped up to handle this is going as smoothly as I could have hoped,” said Mr. Cullen. “It’s been fantastic, and aside from the volume, I couldn’t ask for more.”