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The Duke will make a speech to be broadcast around the world on Thursday, demonstrating his determination to help save endangered African wildlife.
While he is expected to be scrupulously careful to avoid criticising the government, the Duke will win praise from leading Conservatives who want Mrs May to do more.
At last year’s general election, the Tories promised action to ban all ivory sales although this pledge has not yet been implemented. The proposal has been criticised by antiques dealers who say it will put them out of business.
However, senior Conservative figures, including Lord Hague, the former foreign secretary, and Owen Paterson, the former environment secretary, are now demanding that ministers act urgently to fulfil their election pledge.
Mr Paterson is calling on the Prime Minister to announce a ban on ivory sales in Britain ahead of a major international conservation summit this month.
On average, an elephant is killed every 15 minutes for its ivory and their population has fallen by almost a third in Africa since 2007.
Tackling the illegal wildlife trade is a long-standing passion for the Duke. He is patron of Tusk, a charity that campaigns for governments to protect African wildlife and help alleviate poverty.
He will address an event in London hosted by the Tusk Trust this week, and his speech will be broadcast live to audiences in Tokyo and Johannesburg.
Speaking last year, the Duke said: “Let us not tell our children the sad tale of how we watched as the last elephants, rhinos and tigers died out, but the inspiring story of how we turned the tide and preserved them for all humanity.”
He has indicated his support for a ban on domestic ivory sales, arguing that the “root of the illegal wildlife trade” is “the demand for products that require the deaths of tens of thousands of these animals every year, pushing them further towards extinction”.
Charlie Mayhew, chief executive of Tusk, said the event this week would give the Duke the opportunity to make the case for global action ahead of a critical international summit later this month.
The CITES conference in South Africa will discuss action to protect endangered species, although it is not yet clear whether a deal on banning domestic ivory markets will be possible.
Mr Mayhew said he hoped this week’s event with the Duke would help “persuade the British government to meet its commitment to close down domestic sales of ivory in this country”.
“The British government can’t avoid it any longer.”
In last year’s election manifesto, the Conservatives promised to “continue to lead the world in stopping the poaching that kills thousands of elephants each year,” and to “press for a total ban on ivory sales”.
A ban has already been introduced on the international trade but it is still legal to buy and sell ivory inside the UK.
Mr Paterson, the former environment secretary, adds to the pressure on ministers to act today.
Writing in The Telegraph, Mr Paterson says: “The decline in Africa’s elephants over the last nine years is utterly shocking. The Great Elephant Census, published in August, found only 352,271 Savannah elephants across the eight countries surveyed. This alarming decline is the direct result of rising demand for ivory.”
Mr Paterson says the “catastrophic decline in the elephant population” means that further steps must now be taken.
“This cannot mean anything other than closing domestic ivory markets. Although international trade in ivory has been outlawed, ivory can still be bought and sold in individual countries,” he said.
“The 2015 Conservative manifesto made a very clear commitment to closing the UK domestic market. The new Secretaries of State, Boris Johnson and Andrea Leadsom, have signalled that they will treat this matter with the seriousness it deserves, and I hope and expect that we will make good on our promise urgently.”
He says it is “vital” for the government to announce a ban on ivory sale in Britain this week, to show leadership to the rest of the world ahead of the CITES Conference in Johannesburg next week.
“A vote by the UK in favour of closing domestic markets would send a very powerful message, and one which would be followed across the globe. Elephants are being killed by human criminals to fuel human greed and human decisions are now needed to protect them.”
Earlier this month, Lord Hague called on his former government colleagues to honour their election pledge “in full”.
Writing in the Telegraph, he said: “The mass killing of some of the world’s most revered animals is an outrage; a manifestation of human selfishness, stupidity and greed.
“In Britain we have done a great deal to prevent it, but now another step is needed. Often we wait five years for a manifesto promise to be fulfilled: the elephants don’t have that much time.”