Additional funds needed to construct the Iter fusion reactor will have to come from within the EU's budget, member states have said.
The French-based machine will prove the concept of harvesting energy from the fusion of hydrogen nuclei - the same process at the heart of the Sun.
Iter has seen its baseline price tag rise dramatically since a consortium of nations green lit the project in 2006.
The extra 1.4bn euros will cover a shortfall in building costs in 2012-13.
After months of protracted negotiations, member states finally made their position clear at an Agriculture and Fish Council meeting on 12 July.
They want the funds to come from a variety of sources within the existing Brussels budget, including from its research budget (the Framework Programme 7).
The EU's executive body, the Commission, had asked member states to inject fresh funds into the project. The Commission and the European Parliament will have further input before the matter is finally resolved.
Iter itself is holding a special Council meeting on 27-28 July. This will define the latest scope and schedule for the project.
Iter is a collaboration between the EU, the US, Russia, Japan, China, India and South Korea. It is the culmination of decades of research.
Its fusion reactions will take place inside a 100-million-degree gas (plasma) suspended in an intense doughnut-shaped magnetic field.
The reactor is designed to produce 500MW of fusion power during pulses of at least 400 seconds. Critically, Iter is expected to demonstrate the principle that it possible to get far more energy out of the process than is used to initiate it.
The original plan was to build the experiment within 10 years for a budget of 5bn euros. But a range of issues, from technical to personnel matters, have conspired to inflate Iter's final price.
Many now expect it to be in the region of 15bn euros; and the total cost of construction for the EU - a major partner in Iter - is put at no less than 7.2bn euros.
EU ministers had tried - and failed - to resolve arguments over where the extra immediate funds should come from at their 26 May Competitiveness Council.
A task force was then established to find a solution to the issue.
Draft conclusions were agreed at a 30 June meeting of a senior EU body known as Coreper, and these were subsequently adopted as an "A point" at the Agriculture and Fish Council meeting this week.
The member states propose that the additional 1.4bn euros required for 2012-2013 should now be taken from a mix of sources within the current EU budget, including from the Framework Programme.
Many scientists have expressed the fear that raiding the 53bn-euro FP7 pot could damage other projects.
At the European Science Open Forum last week in Turin, Italy, Prof Helga Nowotny, the new president of the European Research Council, again voiced her concern at the impact rising Iter costs could have on other types of European scientific activity.
A spokesperson for the UK government told BBC News: "We are pleased to note the conclusions call for important improvements to the financial management and oversight of the project.
"We agree with the view expressed in the conclusions that the additional financial commitments for Iter for 2012 and 2013 should come from a mix of sources within the current EU budget 2007-2013."