The recent emergence of a coronavirus (SARS-CoV-2), first identified in the Chinese city of Wuhan in December 2019, has had major public health and economic consequences. Although 61,888 confirmed cases were reported in Brazil by 28 April 2020, little is known about the SARS-CoV-2 epidemic in this country. To better understand the recent epidemic in the second most populous state in southeast Brazil - Minas Gerais (MG) - we sequenced 40 complete SARS-CoV-2 genomes from MG cases and examined epidemiological data from three Brazilian states. Both the genome analyses and the geographical distribution of reported cases indicate for multiple independent introductions into MG. Epidemiological estimates of the reproductive number (R) using different data sources and theoretical assumptions suggest the potential for sustained virus transmission despite a reduction in R from the first reported case to the end of April 2020. The estimated date of SARS-CoV-2 introduction into Brazil was consistent with epidemiological data from the first case of a returned traveller from Lombardy, Italy. These findings highlight the nature of the COVID-19 epidemic in MG and reinforce the need for real-time and continued genomic surveillance strategies to better understand and prepare for the epidemic spread of emerging viral pathogens..