Balfour Beatty announced to the stock exchange last Friday that it was pulling out of negotiations to buy the £2bn-turnover business.
The company, which was tipped by Building in March as the frontrunner in the race to buy Jones, has been in exclusive talks with the firm for three months. The asking price was thought to be around the £200m mark.
A Balfour spokesperson said depressed stock market conditions, which have caused the its share price to drop by one-quarter this year, led to it ending the talks.
The spokesperson said: "It's been frustrating. These kind of opportunities don't come along very often. Jones was a very substantial business."
The spokesperson added that Jones would have been a good fit for Balfour. He said: "Jones was a very similar business to what we have in the UK in terms of size and composition."
A statement by the group added: "Current equity market conditions have led the board to conclude that it is not appropriate to raise equity at this time."
Despite the firm's disappointment, its share price jumped 19.5p after the announcement.
Analysts praised the move, even though the bid had cost Balfour £9m in due diligence costs. KBC Peel Hunt analyst Stephen Rawlinson said: "I am happy, in the sense that it would have been easy to have gone ahead with the deal and said how great it was."
City sources said the £9m spend was not wasted. One remarked: "They have learned more about the US market and Jones' clients, including the US government. It will give them an edge for the future."
Balfour was one of three UK groups tipped to buy Jones, which came on the market after Holzmann collapsed in March. Bovis Lend Lease and Amec were linked to the firm, seen as Holzmann's most valuable asset, but Amec immediately denied that it was in the running. The group said it was focused on integrating Spie, its recently purchased French contractor, into its operation.
Jones was founded in 1890 and was bought by Holzmann in 1979.
It has more 9000 staff.