Most hotels, pubs and other businesses checked in the first fortnight of the new English workplace smoking ban were following the law, say officials.
People caught illegally smoking could be fined up to £200
Enforcement officers visited 88,899 premises in early July - and say 97% had a ban in place.
However, councils issued 142 written warnings to employers and licensees - and one defiant smoker was issued with a fixed penalty fine.
Charity Action on Smoking and Health hailed a "fantastic response".
The figures are comparable with the reaction in Scotland and Ireland, which introduced their own bans earlier than in England.
The new restriction, which began on 1 July, covers all workplaces in England, and was designed to offer protection from passive tobacco smoke.
Even cars, lorries and vans operated by businesses were classed as workplaces and smoking forbidden.
Public Health Minister Dawn Primarolo said: "We predicted that it would be largely self-enforcing based on experience elsewhere, and the fact that three quarters of the public supported the move.
"All the signs are that businesses and the public have taken the new law in their stride."
One area which still lags behind slightly is the placement of signs - businesses are expected to put up conspicuous no smoking signs, and 79% of those inspected in July had done so.
The government set up a phone line for the public to report places and incidents in which the new law was being broken.
The "compliance line" received 2,342 calls in the first four weeks, and 606 were passed on to local authorities to follow up.
There are signs that the numbers of calls is starting drop away, with fewer than 400 received in the last week of July.
Amanda Sandford, for Action on Smoking and Health (ASH), said that she had not expected much resistance to the change in law.
"As we predicted, the people of England have welcomed the smokefree law wholeheartedly. It has been largely self-enforcing because it is what the vast majority of people want.
"The smokefree law is working as intended: it is protecting people from toxic tobacco smoke and enabling people to work and socialise, free from the worst form of indoor air pollution."
However, Simon Clark, director of the smokers' lobby group Forest, said the high rate of compliance bore no relation to how popular the ban was.
He said: "It just means that most people are fearful of breaking the law.
"There is still a great deal of unhappiness at the extent of the ban.
"It is far too early to assess the long-term consequences, but we predict that hundreds of pubs will close and the ban will have a negative impact in many places where the local pub or club is the heart of the community."