Background and Objectives: Smartphone use has been rapidly increasing worldwide, which has brought possible smartphone addiction into the focus of research. In order to identify potential smartphone addicts, several scales were developed to assess smartphone addiction. Among them, the Smartphone Addiction Scale was frequently used. The study aimed to test the reliability and validity of the Serbian version of the SAS-SV and estimate smartphone addiction prevalence among medical students. Materials and Methods: The study was conducted in December 2018 on a convenience sample of 323 third-year medical students. The cross-cultural adaptation was performed following the well-established guidelines for cross-cultural adaptation of self-reported measures. Exploratory factor analysis was used to examine the structure of the questionnaire. Factor extraction was performed by principal component analysis with Varimax rotation. For test–retest reliability, students completed the questionnaire twice within seven days. Results: The Serbian version of the SAS-SV showed good internal consistency (Cronbach’s alpha = 0.89) and excellent reliability for test–retest scores (ICC = 0.94, 95% CI = 0.92–0.96). Factor analysis supported the extraction of one factor, which explained 51.538% of the variance. To explore convergent validity furthermore, the SAS-SV was correlated with time indicators of smartphone use. According to cut-off values for the SAS-SV score, 19.5% of students could be regarded as “addicted”, and often spent more time on smartphones and social networks on working days and weekends than “not addicted” students. Conclusions: The Serbian version of the SAS-SV is a reliable and valid instrument for detecting smartphone addiction among university students. Further research on this issue is encouraged to enable a better understanding of this ever-increasing public health issue.