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Bitcoin, the virtual currency that was once the talk of the financial world, has been taking a beating over the last year with the price tumbling downward.

Now two of the biggest boosters of the virtual currency, Cameron and Tyler Winklevoss, are trying to firm up support by creating the first regulated Bitcoin exchange for American customers — what they are calling the Nasdaq of Bitcoin.

The brothers, who received $65 million in Facebook shares and cash in 2008 after jousting with its founder, Mark Zuckerberg, have hired engineers from top hedge funds, enlisted a bank and engaged regulators with the aim of opening their exchange — named Gemini, Latin for twins — in the coming months.

The exchange, which the twins have financed themselves, is a risky bet, given that the virtual currency industry has been a target of hackers and has faced existential questions about its legitimacy. But the brothers are betting that the currency will be able to rise again if it follows the same playbook as the more established financial industry.

“Right now we have to build the infrastructure,” Tyler Winklevoss said. “You have to walk before you run.”

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Since being brought into existence in 2009, by a creator going by the name of Satoshi Nakamoto, Bitcoin has become a technology and financial industry phenomenon. Many major Bitcoin companies, however, were founded by people with little previous financial experience. Bitcoins themselves are stored on a decentralized database run by the currency’s users, and can be bought and sold by anybody.

The twins have a personal interest in seeing Bitcoin succeed. They amassed a small fortune of Bitcoins, starting in 2012, which has declined in value recently. They also have been working with regulators since 2013 to create the first exchange-traded fund holding Bitcoins, for which they are awaiting approval.

They are part of a broader group of wealthy investors and entrepreneurs who have been staking their reputation on the belief that the Bitcoin technology will rise to become more than just a speculative bubble, and will provide new ways of transferring and holding money.

This week, an American Bitcoin company, Coinbase, a kind of retail brokerage firm, announced a $75 million financing round — the biggest ever for a Bitcoin start-up — with backing from the New York Stock Exchange and the Spanish bank BBVA.

But exchanges, where traders can meet to buy and sell Bitcoins for dollars and euros, have proved to be the biggest vulnerability for Bitcoin.

The first major Bitcoin exchange, Mt. Gox in Japan, lost hundreds of millions of dollars and went bankrupt last year. Earlier this month, a security breach at another prominent exchange in Europe, Bitstamp, was the latest reminder of the risks, and helped push the price of a Bitcoin below $200 from a peak above $1,200 in late 2013.

The Winklevosses themselves were swept up in the scandals when the chief executive of a Bitcoin company they had invested in was arrested on charges related to money laundering in early 2014 — a result of activities that happened before they invested.

The twins say none of this has dented their faith in the promise of the technology — they say they continue to hold every Bitcoin they ever purchased — and underscores why a reliable, regulated exchange is needed.

“The A Team wasn’t there,” Tyler Winklevoss said. “There was a problem here and it needs to be solved.”

The Gemini staff is working at a few rows of desks in the Winklevoss Capital offices near Madison Square Park in Manhattan, which has dry-erase boards on the walls covered in math equations and strategic scribbles, and the requisite bean bag chair.

Their chief compliance officer, Michael Breu, was previously at the hedge fund giant Bridgewater Associates, where he was head of information security in the research department. At Gemini, Mr. Breu works closely with the chief security officer, Cem Paya, who previously held the same position at the apartment rental site Airbnb.

They and a staff of a dozen others have been creating Gemini’s security infrastructure and trading engine from scratch, and already have a test model of the exchange running. They are planning to be ready to open the exchange as soon as they win regulatory approval from New York state’s top financial regulator, Benjamin M. Lawsky, the superintendent of the state’s Department of Financial Services.

His office has been leading the effort to regulate virtual currencies in the United States, and is introducing what Mr. Lawsky has referred to as a BitLicense for virtual currency companies. He said recently that he hoped to approve the first companies early this year. A person briefed on the matter confirmed that the office was holding discussions on the Winklevoss effort.

The twins have brought on a leading law firm on financial regulations, Katten Muchin Rosenman, to help win regulatory approval.

Their close cooperation with regulators has also helped them win the thing that has proved the most elusive for Bitcoin companies — a bank account with an American bank. According to documents viewed by The New York Times, on condition that only their general outline be described in this article, the Winklevosses have an agreement with a bank chartered in New York to handle the dollars moving in and out of customer accounts.

Cooperation with regulators has divided the virtual currency world. Bitcoin was founded, in part, with the intention of creating a currency outside the control of governments. The twins have placed themselves firmly in the camp of those who believe that Bitcoin will survive only if it has regulatory oversight.

“Our philosophy is to ask for permission, not forgiveness,” Cameron Winklevoss said.

The biggest Bitcoin exchanges today are all overseas, in China, Hong Kong, London and Eastern Europe, which has not helped instill confidence in some American users.

Most of the exchanges were started at an earlier point in Bitcoin’s development when there was less focus on security and regulations. Bitcoins are particularly vulnerable to hackers because all that someone needs to spend the money in a Bitcoin account is the password, or private key.

The security experts hired by the Winklevoss twins have, like many Bitcoin companies, been focusing on ways to keep the private keys — a mix of letters and numbers — in multiple locations where they are offline and physically guarded.

For the actual trading software, the twins brought on a programmer from the hedge fund Two Sigma, who is building a system that can be used by both small investors and institutional firms that want direct access to the trading system.

The twins are also still moving forward with their exchange-traded fund, set to trade on the Nasdaq stock market under the ticker COIN when approved.

All of these ventures could amount to nothing if the recent decline in confidence continues.

The last major Bitcoin start-up the brothers backed, BitInstant, collapsed in the summer of 2013 before the founder was arrested. After seeing the mistakes made by others, the twins said they wanted to do things on their own.

“This time around we are betting on ourselves,” Tyler Winklevoss said.

Jessica Silver-Greenberg contributed reporting.