Apartment in Copenhagen for rent?

Dear all,

My name is Tim and I'm from The Netherlands. Me and my girlfriend are moving to Copenhagen within a month to live and work there for about 7 months (for an intergovernmental organization). We're looking for a decent apartment close to the city centre (max. 4 km form Tivoli). Maximum price spending on rent will be around a thousand euros. Does anyone have a place available or does anyone have any suggestions/tips.

So far the only places I've found we're rented out by scams, who were asking me to transfer money before I would have even seen the place. I am not looking for that:)

Thanks in advance everyone!

Tim