The relationship between SARS-CoV-2 viral load and infectiousness is poorly known. Using data from a cohort of cases and high-risk contacts, we reconstructed viral load at the time of contact and inferred the probability of infection. The effect of viral load was larger in household contacts than in non-household contacts, with a transmission probability as large as 48% when the viral load was greater than 1010 copies per mL. The transmission probability peaked at symptom onset, with a mean probability of transmission of 29%, with large individual variations. The model also projects the effects of variants on disease transmission. Based on the current knowledge that viral load is increased by two- to eightfold with variants of concern and assuming no changes in the pattern of contacts across variants, the model predicts that larger viral load levels could lead to a relative increase in the probability of transmission of 24% to 58% in household contacts, and of 15% to 39% in non-household contacts.