It went into effect on April 1 after a vetting and filing process of around nine months.
Rountree said he was “very honoured to have been awarded a Royal Warrant as specialist art dealer” to the duke.
Warrants are granted to companies or people who have regularly supplied goods or services to the queen, Duke of Edinburgh or Prince of Wales.
The grantee has the permission and responsibility for the display of the relevant Royal Arms in connection with the business for up to five years.
“It is a testament to the traditional core values of Rountree Tryon that started with the Hon Aylmer Tryon in 1959 and that remains strong within the company ethos,” Rountree added.
“However, I also feel that the brilliance of the gallery staff over the years has really helped to earn this exceptional distinction for the company.”
The gallery, with locations in Petworth and London, specialises in maritime, wildlife, sporting and topographical art from the 18th century to the present day.
It has an exhibition of recent acquisitions running currently at its Petworth base until April 21, including works by artists Alfred Munnings and Archibald Thorburn.