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."