Kostunica told parliament early this month that not enough had been done to make the joint state function properly -- something he intended to change.
"When the [new] Serbian government is formed, we will engage ourselves and in all responsibility will start implementing all our obligations that follow from the Constitutional charter, as well as from our democratic and European orientation," he said. "The government will do everything within its powers to give a practical meaning and to functionally strengthen the state union of Serbia and Montenegro."
In the same speech, Kostunica also proposed the division, or cantonization, of Kosovo along ethnic lines. Some analysts interpreted this as a sign that the moderate nationalist leader may be more prepared to lose at least part of Kosovo than he is to accept Montenegrin independence.
Drasko Djuranovic from Montenegro's independent "Monitor" weekly describes Kostunica as a pragmatic man who might accept that Serbia may never regain control of UN-administered Kosovo. Montenegro, meanwhile, remains a key element of the Serb nationalist ideal.
"Montenegro is a key element in that dream. That means that if you have access to the sea, and if you have Montenegro on your side, that dream of Great Serbia, or Greater Serbia, or whatever you will call it, will continue. Without Montenegro, that dream will definitely collapse," Djuranovic said.
Kostunica's message could not have been welcome for Montenegro's pro-independence leaders. Speaking on the second anniversary of the union agreement earlier this month (14 March), Prime Minister Milo Djukanovic dutifully reiterated Montenegro's commitment to what he described as its “international obligations.”
"Today, we are continuing to implement the Belgrade agreement. Montenegro will continue doing this in a responsible manner, first and foremost by conducting itself responsibly toward what it considers its international obligations."
But he also said that Montenegro is ready to discuss with Serbia a "more rational solution," and that if such a solution is not found, Montenegro will "wait for its chance to test the will of its citizens" on state sovereignty.
Serbian and Montenegrin leaders differ on a long list of issues -- from the powers of the central institutions, to cooperation with the UN war crimes tribunal and priorities in foreign relations.
If maintaining the state union proves too difficult, Djuranovic says, the two sides will have to choose between an agreeable split and a referendum.
"Podgorica is trying to persuade Belgrade [to consider] some kind of dissolution of the union according to the Czechoslovak model," he said.
A referendum held without first securing national consensus on the issue could be politically risky for Montenegro's government. And that, says Djuranovic, may tempt Belgrade to force Podgorica's hand.
"Belgrade will push Podgorica to a referendum, because they are very aware that some kind of a referendum in Podgorica will be very, very difficult to organize and it will be a small majority in favor of independence. And that means that probably the international community will never recognize that kind of referendum."
Analysts say the European Union may frown upon Kostunica's nationalistic stance, but that does not mean it would welcome an independence referendum.
The EU earlier this week reiterated that in order to move toward European integration, Serbia and Montenegro must have strong central institutions and a single market. The EU is expected within weeks to release its assessment on whether Serbia-Montenegro is ready to start negotiations on an association agreement with the bloc.
Kostunica and Djukanovic, meeting yesterday for the first time since Kostunica's election, said in a statement their governments had "common goals" and would step up work toward European integration.
(RFE/RL's South Slavic and Albanian Languages Service contributed to this report.)