Theresa May is trying to persuade the 10 DUP MPs to enter into a formal coalition agreement with the Tories, as opposed to a less formal "confidence and supply" arrangement, ministers tell me.
The prime minister has sent a team of officials, led by her chief whip, Gavin Williamson, to Belfast to negotiate the details of an alliance with the DUP.
"A coalition would be much better than a looser alliance", one senior minister said. "We don't want the DUP demanding money for this or that project they fancy every time we need them to support us in a vote. That would be deeply unstable".
"Don't under-estimate the importance of doing the DUP deal" another of Mrs May's colleagues told me. "Without it, I don't see how we govern with any confidence".
Or to put it another way, the deal with the DUP represents life or death both for the new government and for Mrs May as PM.
Mrs May's hope is that in any coalition, the DUP would be satisfied with ministerial jobs at just below cabinet rank. It is not clear whether the DUP would insist on being represented in the cabinet.
Sources close to the prime minister say that she understands the concerns of liberal Tories like Ruth Davidson about the possible price of a deal with the DUP, and she will not offer them any promises to reduce the rights of the lesbian, gay, bisexual, trangender or intersexual community, or to reduce womens' rights to abortion.
Promises likely to be made by Mr Williamson and the prime minister are likely to concentrate on the Northern Ireland implications of Brexit.
In particular, they will pledge to try to ensure that customs and immigration checks will not be introduce between Northern Ireland and the Republic, and that there will be no new restrictions on free movement between Northern Ireland and the British mainland.
There are concerns that the rest of the EU may insist on customs checks between Northern Ireland and the Republic.
Mrs May is also expected to offer to invest in Northern Ireland infrastructure.
She is working today on finalising her cabinet.