Since its outbreak, coronavirus disease 2019 (COVID-19) has rapidly spread throughout the world. The Serbian government declared a state of emergency on 15 March, 2020, implementing some of Europe’s strictest measures to combat the pandemic. The aim of this study was to determine the impact of the COVID-19 epidemic on the mental health of the general adult Serbian population and to identify associated factors during the state of emergency and lockdown. Data were collected with a snowball sampling method between 23 March and 25 April, 2020, by using an online questionnaire. Multiple ordinal regression was performed to establish the associations between socio-demographic characteristics, self-estimated health status, and depression, anxiety, and stress. Out of 1057 participants included in the study, 28.9%, 36.9%, and 38.1% reported moderate to severe depression, anxiety, and stress symptoms. Uneasiness related to COVID-19 news, the feeling of helplessness, likeliness of impending death, and presence of COVID-19 symptoms were associated with higher depression, anxiety, and stress scores. Current smoking status was associated with a higher risk of depression and stress. Students had a significantly higher level of depression and stress, while older age was protective against anxiety and stress. Higher socioeconomic status was significantly associated with lower levels of depression, anxiety, and stress.