The vast majority of pubs and clubs complied with the English smoking ban during its first month, new figures from the Department of Health show.
However, around one in 12 pubs and clubs did not display the correct signage, although this was well ahead of some premises where up to one in five did not comply.
Data for the whole of July shows a 97.3% compliance rate among the 16,807 licensed premises - excluding restaurants - inspected.
Compliance was almost identical for restaurants (97.8%), venues with accommodation (97.2%) and other premises (97.1%).
The rate of compliance for signage was 91.3% for pubs and clubs, compared to 86.7% for venues with accommodation, 86.6% for restaurants and 79.5% for other premises.
There were 312 written warnings issued to premises failing to prevent smoking, and one court hearing.
More than 3,000 written warnings were issued to all premises for failing to display the correct signage. Two premises received fixed penalty notices for the offence.
Referring to the high level of compliance among pubs, Federation of Licensed Victuallers Associations (FLVA) chief executive Tony Payne said: "I'm pleased with these figures. It shows that pubs and clubs are honouring the smoking ban.
"Although we did our part, we should also credit the trade papers for making people aware in advance."
Payne suggested that the non-compliance with the signage requirements among some venues was because the Government only sent two signs to premises before the ban. He said: "There are bound to be some people who weren't in the loop, who didn't get signs sent to them and didn't have the awareness to ask for more signs.
I don't think it's deliberate."
variations in compliance
Pubs in north-east England were most compliant with the smoking ban - while those in Yorkshire and Humberside were the most likely to breach the law.
Figures show there was little regional variation overall. As for compliance among premises, levels varied from 95.3% in Yorkshire and Humberside to 99.2% in the North East.
Regional variations were more evident around signage, ranging from 79.3% in Yorkshire and Humberside to 98% in north-east England.
air tests in somerset pubs reveal drop in carbon monoxide levels
The smoking ban has been shown to reduce levels of carbon monoxide (CO) in pubs by two thirds.
Air was tested at four Somerset pubs in May, and re-tested in August. Tests in August revealed an average 64% drop in CO levels.
North Somerset Council used an electrochemical carbon monoxide monitor to measure the air every five minutes over three days. CO can be created by cigarette smoke and other things like boiler emissions.
Clevedon's Crab Apple registered the biggest fall in CO (87%), followed by the Anchor Inn, Easton-in-Gordano (73%) and the Blue Flame, Nailsea (71%). The Royal Oak in Nailsea saw a smaller drop of 33%.
A council spokeswoman said levels of CO reduction can vary due to the size of the pub, ventilation and how smoky it was previously.
Mick Davidson, freehold owner of the Blue Flame, said: "The pub is definitely cleaner and fresher. I've found it so much nicer." He also expressed fear that the ban would hit trade in the winter - and said the open fire used in cold weather will see smoke return inside.
The council's executive member for the environment Carl Francis Pester said: "It is excellent to see the positive impact the smoke-free laws are having on work places, particularly pubs."