Iain Cornish, chief executive of the Yorkshire, said that there would be no windfall for members because economic conditions were tougher than at the time of previous building society deals.
The merger is likely to be interpreted as a rescue deal for Chelsea, currently the UK's fifth largest building society , whose new chairman Stuart Bernau has been reviewing the future of the business and whether it should stay independent.
In August, it reported a half-year loss of £26m after setting aside £41m to cover the cost of mortgage frauds.
In 2008, it reported a full-year loss of £39m, the largest yet recorded by a building society, brought on by losing £44m which had been invested in two failed Icelandic banks.
"Since the onset of the 'credit crunch' Chelsea has experienced a period of disappointing financial results arising from a range of factors," the Chelsea said.
"[The merger] will create a larger building society with greater capital strength and deeper financial resources than Chelsea.
"Without the merger... your board believes that Chelsea would not be able to continue to provide the same range of products or good value pricing to its members," it added.
The Yorkshire said it had also recorded a loss, of £22m, for the first half of 2009.
The aim is for the merger to have taken place by April next year.
There will be an unspecified number of job losses when the merged society cuts duplicated overhead costs.
"The transaction is expected to deliver significant annual savings which are anticipated to be fully realised within 18 months after the merger, creating a more efficient organisation," the Chelsea said.
Anyone who has savings with both the Chelsea and the Yorkshire will retain their separate £50,000 cover for each society under the Financial Services Compensation Scheme (FSCS), but that dual cover will expire on 30 December 2010.
After that point, savers with money in both societies will only receive £50,000 of cover under the FSCS for their total savings in both the Yorkshire and Chelsea.
Members of both societies will receive voting packs in the next few weeks.
No public money is involved in the deal and there will be no bonuses for officers or directors of either society as a result of the merger going through.
Unlike banks and other stock-market quoted companies, building societies, which are owned by their members, have a limited ability to replenish their reserves if they are drained by losses.
Since the autumn of 2008 there has been a flurry of takeovers of small building societies to rescue them from problems brought on by the financial crisis.
In September last year, the Nationwide agreed to stage a rescue takeover of both the Cheshire and the Derbyshire.
The Yorkshire then took over the Barnsley building society, while the Skipton took over the Scarborough.
In March this year, the Dunfermline building society collapsed, to be taken over by the Nationwide.
This was swiftly followed in June by emergency action by the authorities to ensure that the loss-making West Bromwich could stay afloat.
Meanwhile one of the UK's largest and strongest building societies, the Britannia, has merged with the Co-op bank.
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