But overall, public attitudes seem to justify the policy of both Labour and Conservative parties, which is to rule out decriminalising or legalising the drug for the foreseeable future.
Previous polls have not asked quite the same questions, but it seems the Independent poll shows not just a reluctance to liberalise the law, but an actual hardening of attitudes, which perhaps corresponds to an increasing intolerance of smoking and drinking, particularly in public.
Last year, while the Liberal Democrats were advocating the decriminalisation of cannabis, Michael Howard, the Home Secretary, moved in the opposite direction, increasing the maximum fines for possession fivefold, to £2,500. His Labour shadow, Jack Straw, equally terrified of tabloid press headlines, has ruled out any move towards legalising or decriminalising.
The poll results are in tune with their caution. Six out of 10 people think cannabis should remain illegal. Just under a quarter think the drug should be legalised completely like cigarettes, while fewer than one person in eight favours the Liberal Democrats option of decriminalising, which would leave its possession as a minor offence subject to a small fine, like speeding or parking.
Attitudes against cannabis harden with age, but the surprising aspect is not how greatly they strengthen, but how little.
Among the 15-24 age group, almost half believe cannabis should remain illegal, rising to three-quarters in the over-55s. The 45-54 age group - which would include those who were students in the late 1960s and early 1970s when cannabis smoking among the young was probably at its most public - are far from liberal now, with almost seven out of 10 against any move towards legalisation.
The question such polls do not answer is whether young hippies have become middle-aged reactionaries, or whether they were never more than a very visible minority. More importantly, the poll cannot tell the politicians whether the fact that a majority of the under-35s favour some legalisation now means that in a generation's time there will be a majority in the country as a whole.
Either way, politicans tend to work on a short timescale; and Mr Howard will be heartened by the degree of support for the current Government policy on cannabis: only one person in five thinks it is too tough, and almost two-thirds think it is either about right or should be harsher. Only one person in seven thinks the law is enforced too strictly, and four in 10 people want harsher enforcement, with more than a third thinking the police have got it about right. But the figures also show that one young person in three has tried cannabis, which might be taken to suggest that for all the enforcement, an illegal substance is surprisingly readily available.
It is interesting how the attitudes to penalties and enforcement differ when the questioners ask people who have tried cannabis themselves, and are therefore more likely to have first-hand experience of what happens. Among this group, six out of 10 think the Government policy is too tough, and almost half thinks the law is enforced too strictly. Only very small percentages regard the law and enforcement as too weak.
Region makes a difference to people's attitude to legalisation or decriminalisation of cannabis. In London there is a majority in favour of relaxing the law, and in Scotland opinion is only narrowly against.
Keith Hellawell, Chief Constable of West Yorkshire and chairman of the Association of Chief Police Officers committee on drugs was surprised at the extent of the poll's findings that penalties and enforcement were too weak.
"When the Home Secretary recently increased the penalties for the possession of cannabis he was criticised in many quarters," he said. Police statistics show that more people are arrested each year in connection with cannabis and other proscribed drugs.
The strong support for legalising cannabis as a prescribed drug is the only area in which a government might feel it is out of sympathy with public opinion when it advocates the status quo. In fact, there has been a historic divide between the Home Office, which sees cannabis and all illegal drugs in terms of problems and as a contributor to crime, and the Health Department, which has an open mind towards the possibility that the licensed use of illegal drugs under medical supervision can be beneficial.
The Department of Health has not ruled out the use of cannabis on prescription after clinical trials, but stresses there is a debate about the long-term effects of cannabis use. Here there may be a compromise solution in the short term. In an written Commons answer last year, the junior Health Minister, John Bowis, pointed out that a drug called nabilone, which contains the main active ingredient of cannabis, is already licensed in the UK for the treatment of nausea in patients with cancer.
"A doctor may prescribe nabilone for other conditions on his own responsibility," he said. A study in today's Lancet suggests nabilone may have the same beneficial effects on multiple sclerosis claimed by some doctors for cannabis. Two doctors reported that nabilone produced a "striking reduction" in severity of muscle spasm and an increased mood of well-being. Other doctors have already been prescribing it, according to the Multiple Sclerosis Society.
One of the strongest arguments against liberalising cannabis in any way is that the long-term effects on health cannot be known, and might only strike the population 20 or 30 years after the change in the law. This is less of a risk if the drug is being administered to people who are already seriously ill, but it will always remain at the back of the mind of any politician responsible for legalising cannabis - even if there is a majority of the population in favour.
The findings of the cannabis survey, conducted for the `Independent' by Harris Research, will be the subject of tomorrow night's televised `Great Pot Debate', chaired by Sheena McDonald, on Channel 4 (8pm).Reuse content