Scottish First Minister Nicola Sturgeon has already warned the new Prime Minister that for Scotland, which voted pro-EU in every region, "Remain means Remain", stressing that while "Theresa May said in her view Brexit means Brexit".
Plaid Cymru leader Leanne Wood said Wales should be given "every penny" of lost European Union funding after Brexit and suggested the country should be allowed to have its own independence referendum if Scotland pulls out of the UK.
Mr Jones, who had not met the former home secretary before, called the meeting "a constructive and positive start to our working relationship".
"I have long argued that the Conservative Party must be a "One Nation" party and we must continue to focus on helping those who feel that society and government isn't on their side".
Ministers banged on the table after the prime minister made her opening remarks at the meeting, which is being used to set out the government's priorities.
"At Cabinet this morning you saw the Prime Minister really encouraging a collective government approach". She said she recognised concerns on the loss of European Union funding and described the meeting as "very constructive".
"I am very honoured and delighted to have been appointed to this role", he said.
Downing Street says arts minister Ed Vaizey, a friend of Cameron, has left the government.
May assembled her closest allies at 10 Downing Street for the first time on Tuesday morning, where she told the likes of Foreign Secretary Boris Johnson, Health Secretary Jeremy Hunt and one-time leadership rival Andrea Leadsom that social reform, not Brexit, should be the hallmark of her tenure. "We look forward to this translating into strong action to address the costs that British steelmakers face, that our overseas competitors don't, such as higher energy costs and business rates".
"The government must also ensure that major infrastructure projects are set in motion in short order and that steel made in the United Kingdom is used to the full in these great British projects".