The US could indict Swiss banks, and possibly even cut them off from the dollar market.
"The bill is the result of pressure from the US following revelations that Swiss banks had helped American account holders to evade taxes.
The US had demanded action by 1 July, but the Swiss parliament summer session ends this week.
The bill will now return to the Senate.
The lower house decided by 126 votes to 67 not to discuss the bill. A second rejection by the lower house would effectively kill the draft law.
The bill would allow Swiss banks to sidestep strict secrecy laws and release information relating to clients' accounts.
It also contained secret clauses requiring the banks to pay an estimated $10bn (£6.4bn) in compensation for lost tax revenue.
Swiss politicians are in no mood to change their laws because Washington tells them to, reports the BBC's Imogen Foulkes from Berne.
The Senate very reluctantly approved the bill last week, after it became clear the US would indict Swiss banks, and possibly even cut them off from the dollar market if it did not go through.
Switzerland has also come under pressure from the EU over the issue.
In January Switzerland's oldest private bank, Wegelin, closed after being indicted and fined $58m by the US authorities after admitting in court to helping American customers to hide more than $1.2bn from the Internal Revenue Service. In 2009, Swiss bank UBS paid $780 million and handed over details of more than 4,000 accounts in order to avoid indictment.