Background: The COVID-19 pandemic has led to major social and economic changes that could impact public mental health. The main aim of the current study was to investigate mental health in Norway during the COVID-19 outbreak (since the first confirmed case on 26 February 2020). Methods: The results are from the first wave of the data collection (1 April–2 June 2020), which took place during the outbreak along with its initial restrictions. A total of 19,372 (11,883 students) people participated in a cross-sectional web-based survey. Results: A total of 21.8% scored above the cut-off for depression and 23.7% for anxiety. Severity of symptoms was associated with the accumulation of risk factors, such as possible/confirmed infection for oneself or one’s family, female/other sex, students, having mental health problems, increased use of tobacco, increased use of alcohol, less exercise, losing one’s job, suffering economic impact and lower education. Conclusions: COVID-19 could have a negative association with public mental health, especially for certain risk groups. Future data-collection waves will provide further insight into the development of symptoms following the pandemic.