Hepatitis C virus (HCV) is present in the host with multiple variants generated by its error prone RNA-dependent RNA polymerase. Little is known about the initial viral diversification and the viral life cycle processes that influence diversity. We studied the diversification of HCV during acute infection in 17 plasma donors, with frequent sampling early in infection. To analyze these data, we developed a new stochastic model of the HCV life cycle. We found that the accumulation of mutations is surprisingly slow: at 30 days, the viral population on average is still 46% identical to its transmitted viral genome. Fitting the model to the sequence data, we estimate the median in vivo viral mutation rate is 2.5×10⁻⁵ mutations per nucleotide per genome replication (range 1.6-6.2×10⁻⁵), about 5-fold lower than previous estimates. To confirm these results we analyzed the frequency of stop codons (N = 10) among all possible non-sense mutation targets (M = 898,335), and found a mutation rate of 2.8-3.2×10⁻⁵, consistent with the estimate from the dynamical model. The slow accumulation of mutations is consistent with slow turnover of infected cells and replication complexes within infected cells. This slow turnover is also inferred from the viral load kinetics. Our estimated mutation rate, which is similar to that of other RNA viruses (e.g., HIV and influenza), is also compatible with the accumulation of substitutions seen in HCV at the population level. Our model identifies the relevant processes (long-lived cells and slow turnover of replication complexes) and parameters involved in determining the rate of HCV diversification.