Bitfinex allows users to trade in several different digital currencies, including bitcoin, and deposit U.S. dollars in their account.
A total of 119,756 bitcoins, worth $70.5 million at today's price, were stolen as a result of a cybersecurity breach.
Bitfinex's decision is likely to disappoint many of the site's users who held assets other than bitcoins, warned Charles Hayter, chief executive and founder of digital currency comparison website CryptoCompare.
"What's disappointing is that the losses seem to be arbitrarily decided with larger operators being offered sweetners to keep them trading - and there is little clarity on BitFinex's company losses," Hayter told CNBC via email.
To compensate users, BitFinex is crediting each account with a digital token that will record how much the customer has lost as a result of the hack. These tokens will either be redeemed in full by the company in the future or they can be exchanged for shares in BitFinex's parent company, iFinex Inc. The statement did not specify any timeframe for the redemption.
"The convertible debt token is a way of kicking the can down the road and finding breathing space for the exchange - it opens up interesting trading possibilities with its junk status as well as a fair few legal ramifications," explained Hayter.
Hayter criticised BitFinex's latest attempt to deal with the situation.
"It's all been desperately scrambled together to give some form of closure - although a lot of their plan has not been fully fleshed out with details thin on the ground," he said.