In a report to the UN Security Council on Wednesday, Ashdown applauded the Republika Srpska authorities for transferring five indictees to the UN court in the last two months, which he said indicated their attitude to the tribunal had changed.
Ashdown added it was too early to say the entity was cooperating fully and that it was natural for experienced observers to remain suspicious.
He said he believed the entity authorities had started accepting that the road to the EU and NATO passed through The Hague, but added there could be no let-up until all the accused were arrested.
The report said that 10 years after the Dayton agreement ended the war in Bosnia, the country was focusing on meeting the terms to enter NATO's Partnership for Peace programme and to begin stabilisation and association negotiations with the EU. The main obstacle to this has been lack of full cooperation with the Hague tribunal on Republika Srpska's part.
NATO has recently rejected Bosnia's application for Partnership for Peace for the second time and will reconsider the application in April. If it is turned down for the third time, the country will no longer be invited to join.
Ashdown said the recent transfers did not constitute full cooperation with the Hague tribunal, and that 10 years after Srebrenica, where Bosnian Serbs killed about 8,000 Muslims, it was important to arrest all those responsible, including Radovan Karadzic and Ratko Mladic.
Ashdown said Bosnia-Herzegovina must meet other international requests as well, including police reorganisation and defence reform, which the Serb entity objects to. He voiced hope 2005 would be the year the country would start becoming part of Euro-Atlantic structures.
Bosnian Security Minister Barisa Colak told the Security Council he expected the EU to give Bosnia credit for its progress and approve launching negotiations on a stabilisation and association agreement in May.
Speaking on behalf of current EU president Luxembourg, Ambassador Jean-Marc Hoscheit said that despite the progress and the meeting of criteria, there remained obstacles like cooperation with the Hague tribunal and police reorganisation. He added that full cooperation with the UN court, notably on the Serb entity's part, remained crucial.
The report also said that Croatian President Stjepan Mesic's re-election earlier this year would make a positive contribution to stability in Bosnia as well as in the entire region.
Ashdown said this was his second to last report to the Security Council because he would step down from the post of High Representative to Bosnia towards the end of the year.