There is basketball, ping pong, football and bingo, as well as daily video shows, and visits from pop bands. "We have all heard about Denmark. It is a very equal society which tries to help people like us. It has a good human rights record and the best education in the world. I would like the chance to start a life here," says Mr Fraije, a Palestinian refugee.
The asylum-seekers come and go as they want, and their needs are answered by a team of jolly Red Cross workers like Hele Kampegaard, who sweeps round the camp in a fur coat, her blonde hair tousled by the icy Zealand wind.
"The families all get their own bathroom," she says, as we pass the entrance gate where two Bosnian Muslims have just arrived, via Germany, to ask for refuge here. An Iraqi businessman in a suit is holding up a Baghdad- Amman bus ticket to prove where he came from.
"Our job is to make sure they are all looked after. They come here because they have heard about our democratic system. We have the highest standards in Europe. They think they can come here then go and build their own countries back home."
The increasing number of asylum-seekers attracted by famous Danish altruism is causing anxiety throughout Danish society, and is now the dominant issue in the Danish debate about European power-sharing.
There are many in Denmark who believe altruism has gone far enough - at least when it comes to "foreigners". The answer, they say, is to accept European immigration rules and standards, by dropping the Danish opt-out from European Union justice policy, after the Amsterdam summit in June.
Others warn that to drop the opt-out, which could only happen after a referendum, would be to fall into the trap of accepting ever further European integration.
EU leaders are increasingly presenting the fight against illegal immigration and international crime as a prime objective, in their attempt to give the union new credibility with "citizens". The so-called "third pillar" of EU government, governing justice policy, is therefore expected to be strengthened in the Amsterdam treaty.
Danes, however, have always been deeply suspicious of giving Europe political powers outside the economic sphere. Furthermore, such a move would threaten Denmark's much-valued human rights and social standards. "What people fear is what this could lead to. It could lead to a European police force - to German police running around Danish soil," said one Danish diplomat.
There are fears that if the government is to win Danes round to giving up one of their precious "opt-outs", won in 1992, they will have to play the race card, by fuelling fears of immigration and saying the EU would provide a defence.
At first it is hard to see why immigration has become an emotive issue here. Black or brown faces are rarely seen. A small, highly regulated state of just 5 million, Denmark is not a country where "illegals" can easily remain hidden and it has always been hard for foreigners to find work here. As one government official said: "Most Danes have never met an immigrant."
It is, perhaps, precisely because of the country's long-standing homogeneity that the growing number of immigrants arriving here is causing such trauma.
Although the numbers are relatively tiny - there were 6,000 asylum-seekers last year, compared with 5,000 in 1995 - there are fears that growing hordes are simply waiting on Europe's outer rim to take advantage of Denmark's generosity.
People are arriving because they are being refused entry by other EU states - particularly Germany. Eleven Somalis, refused asylum in Germany, were recently discovered in a car crossing the German-Danish border. Denmark granted all Bosnians asylum during the conflict, whereas Germany only granted temporary protection. In the spring, Bonn is expected to forcibly return its Bosnians, and Denmark fears many might try to come here.
Publicity given to such cases has played into the hands of the Danish far-right. Permissive Danish laws on free speech allows neo-Nazi groups here to operate with relative freedom. To many Danes, it is starting to look attractive to be inside the EU's defensive "ring fence."
A new set of instruments is being agreed in Brussels to tackle the threat immigrants are deemed to pose to the union. The buzz-word among European immigration watchers is "burden-sharing" - a euphemism for setting up asylum quotas.
Danish authorities fear that if they are not a part of the EU system, Denmark will be forced to take more of the "burden" of refugees than is fair, as asylum-seekers head for shelter here.