Finding informative COVID-19 posts in a stream of tweets is very useful to monitor health-related updates. Prior work focused on a balanced data setup and on English, but informative tweets are rare, and English is only one of the many languages spoken in the world. In this work, we introduce a new dataset of 5,000 tweets for finding informative COVID-19 tweets for Danish. In contrast to prior work, which balances the label distribution, we model the problem by keeping its natural distribution. We examine how well a simple probabilistic model and a convolutional neural network (CNN) perform on this task. We find a weighted CNN to work well but it is sensitive to embedding and hyperparameter choices. We hope the contributed dataset is a starting point for further work in this direction.