27 September 2012 Last updated at 12:32 ET
The meeting took place on the sidelines of the United Nations General Assembly
The UK is legally obliged to extradite Julian Assange to Sweden, Foreign Secretary William Hague has told his Ecuadorean counterpart.
Mr Hague held talks with Ricardo Patino in New York.
The Wikileaks founder, who is staying at Ecuador's London embassy, is wanted in Sweden for questioning over sexual assault claims, which he denies.
Ecuador was sheltering him under the concept of "diplomatic asylum" which did not exist in UK law, Mr Hague said.
Mr Hague added that UK extradition law included wide human rights safeguards and "requested the government of Ecuador to study these provisions closely in considering the way ahead".
The meeting took place on the sidelines of the United Nations General Assembly.
The Foreign Secretary's spokesman added: "Both ministers agreed that they were committed to the search for a diplomatic solution to Mr Assange's case.
"They were willing to meet again at this level in due course to continue these exchanges."
Mr Assange, 40, has now been at Ecuador's embassy for 100 days.
In 2010, two women accused him of committing sexual offences against them while he was in Stockholm to give a lecture.
He says the sex was consensual and he regards the case as politically motivated as he fears he could be handed over the US and put on trial over Wikileaks.
His website has published leaked sensitive diplomatic cables involving various countries, including the US.
The talks between Mr Hague and Mr Patino came after Mr Assange addressed a fringe meeting at the General Assembly via videolink from Ecuador's London embassy.
The UK is legally obliged to extradite Julian Assange to Sweden, Foreign Secretary William Hague has told his Ecuadorean counterpart.
Mr Hague held talks with Ricardo Patino in New York.
The Wikileaks founder, who is staying at Ecuador's London embassy, is wanted in Sweden for questioning over sexual assault claims, which he denies.
Ecuador was sheltering him under the concept of "diplomatic asylum" which did not exist in UK law, Mr Hague said.
Mr Hague added that UK extradition law included wide human rights safeguards and "requested the government of Ecuador to study these provisions closely in considering the way ahead".
The meeting took place on the sidelines of the United Nations General Assembly.
The Foreign Secretary's spokesman added: "Both ministers agreed that they were committed to the search for a diplomatic solution to Mr Assange's case.
"They were willing to meet again at this level in due course to continue these exchanges."
Mr Assange, 40, has now been at Ecuador's embassy for 100 days.
In 2010, two women accused him of committing sexual offences against them while he was in Stockholm to give a lecture.
He says the sex was consensual and he regards the case as politically motivated as he fears he could be handed over the US and put on trial over Wikileaks.
His website has published leaked sensitive diplomatic cables involving various countries, including the US.
The talks between Mr Hague and Mr Patino came after Mr Assange addressed a fringe meeting at the General Assembly via videolink from Ecuador's London embassy.