Ukip support has plunged back from its high point of the European elections, giving a boost to all the established Westminster parties and pushing the Tories into the lead, according to a new Guardian/ICM poll.

Nigel Farage's party drops back into single figures to stand on 9%, down seven points from last month's score of 16%. The Tories pick up the lion's share of these deserters, gaining three points to climb to 34%, enough to put them a single point ahead of Labour, even though the opposition also edges up one to 33%.

The Liberal Democrats also climb two, to 12%.

As David Cameron prepares to shuffle his ministerial deck, he will be relieved to have the edge, but the one-point margin between the Conservative and Labour parties is a statistical dead heat. This is the third straight month in the 30-year polling series in which the race has been virtually tied, after a statistically insignificant lead for the Tories in May was followed by a similarly tiny advantage for Labour in June.

Ten months ahead of a general election, Martin Boon, director at ICM research, said: "We used to talk about parties getting themselves through the 'magic' 40% threshold before they would be in serious contention to win, but less than a year before a general election, both the big parties are currently struggling to get themselves into the middle 30s, which, of course, only the Tories managed in 2010."

Although the collapse in Ukip support is sharp this month, Boon suggests it is "pretty typical" of what happens to them as the spotlight cast by mid-term elections recedes. "This time last year," he says, "Ukip also dropped to a similar extent, from 18% in the ICM/Guardian May 2013 poll to 12% the following June." The more striking thing, as he sees it, is the failure of the declining protest vote to split decisively in any direction. Too many voters, he says, see "no reason to commit just yet", instead "holding off, perhaps until the end of this year, to see whether they can really feel the recovery in their pockets or not by that point".

While many recent polls point to a tight race next year, in most, Labour retains the edge. ICM, which has the best record in recent general elections, differs from many others in continuing to conduct its surveys of voting intention over the telephone as opposed to online. It also makes a distinctive adjustment to deal with voters who are happy to report how they voted last time, but are less forthcoming about what they will do in the future. It assumes that many such voters "return home" in future elections, which in the past has been a useful way to identify "shy Tories" at times when the Conservatives have been unfashionable.

In Monday's data, however, ICM's adjustments do not much change the relative standing of the main two parties – Labour and the Conservatives would both be on 35% without the adjustment, although the Liberal Democrats would fare worse – they stand at just 9% before the assumption about some current deserters returning to the fold is applied.

Although Cameron is reported to be planning to promote several women ministers to give his government a more "inclusive" feel this week, there is no marked gender gap in the latest data. The main parties are in a virtual dead heat among both men and women, a contrast with several surveys over the last couple of years, when substantial deficits for the Tories were underpinned by evidence of a particular problem with women. There is little evidence, either, of social class exerting much of a hold – Conservative support is not much lower among the lowest DE occupational grade (30%) than it is among AB professionals (34%).

The fault-lines that do emerge as important concern age and region. The over-65s are hugely more likely to support the Conservatives than anyone else, with 52% backing the Tories compared with less than 30% across most of the rest of the age range. There is also a marked contrast in Tory support between Scotland (21%), Wales (18%) and the north as a whole (21%), a very different picture from the Midlands, where Cameron's party stands on 46%.

ICM research interviewed a random sample of 1,000 adults aged 18+ by telephone on 11-13 July 2014. Interviews were conducted across the country and the results have been weighted to the profile of all adults. ICM is a member of the British Polling Council and abides by its rules.