"Australia has called for a free trade deal with Britain following its exit from the European Union.
In a phone call to the new PM, her Australian counterpart Malcolm Turnbull said he urgently wanted to open up trading between the two countries.
Liam Fox, the new international trade secretary, said he was already "scoping about a dozen free trade deals".
But Britain cannot sign any deals while it is still an EU member. Mrs May said: "I have been very clear that this government will make a success of our exit from the European Union.
"One of the ways we will do this is by embracing the opportunities to strike free trade deals with our partners across the globe. It is very encouraging that one of our closest international partners is already seeking to establish just such a deal."
"This shows that we can make Brexit work for Britain," she added.
Dr Fox, a prominent Brexit campaigner, said numerous non-EU countries had already asked Britain for a trade deal and he was "scoping about a dozen...to be ready for when we leave".
It comes amid reports he is preparing to fly to the United States next week for talks.
In April, President Barack Obama warned the UK it would go to the "back of the queue" for trade deals with the US if it voted to leave the EU. Following the referendum, he said the UK's decision to leave raised "longer-term concerns about global growth"."...
[Ed. note: You people at the BBC are pathetic. You hang on every scripted word the media creation/Wall St. puppet Obama says. Get a life. He'll be gone in 6 months. The US doesn't have monarchs. It's not likely the UK will be officially out of the EU in less than 6 months nor that TTIP will be passed in both Europe and the US in less than 6 months--if ever. Lame duck presidents and congresses usually do hateful things on their way out the door. Obama could, for example, sign a bunch of Executive Orders. But they can all be canceled by the next president.]
(continuing): "Mr Fox told the Sunday Times: "We've already had a number of countries saying, 'We'd love to do a trade deal with the world's fifth biggest economy without having to deal with the other 27 members of the EU.'"
Meanwhile, the new minister in charge of Brexit has said the UK should be able to formally trigger its departure from the EU "before or by the start of next year".
David Davis called for a "brisk but measured" approach, with a likely exit from the EU around December 2018.
He told the Mail on Sunday "we will get a generous settlement for EU migrants here now and a generous settlement for British citizens in the EU" but that tighter rules may have to be brought in for those coming to the UK before Brexit happens....
On Friday, Mrs May told Scottish First Minister Nicola Sturgeon she would not trigger Article 50 to leave the EU before getting UK-wide agreement and she was "willing to listen to options" on Scotland's future after it voted to Remain."
......."BBC correspondent Phil Mercer, in Sydney"
"Britain is Australia's seventh largest trading partner, and is second only to the United States when it comes to direct foreign investment down under.
Australian Prime Minister Malcolm Turnbull said a free trade agreement with the UK was a priority, although such treaties are complicated and can be time-consuming.
Australia's recent trade deal with China, for example, took a decade to negotiate.
Mr Turnbull has said Canberra could also team up with New Zealand to strike new commercial and immigration accords with the UK following its decision to leave the EU."