Previous studies have found mixed evidence regarding the effects of the minimum wage on training levels. This paper exploits a discontinuity in the minimum wage received by apprentices in the United Kingdom to examine this question. Workers aged 19-20 receive a substantial increase in the minimum wage after one year on an apprenticeship, whereas workers aged under 19 do not experience a change in the minimum wage at this point. Using data from the Apprenticeship Pay Survey, regression discontinuity design estimates suggest that the increase in the minimum wage has no overall effect on training among 19-20 year-olds. However, among firms that are compliant with the minimum wage legislation, the minimum wage reduces training by 11-23%. Since relatively few employers pay exactly the minimum wage, this implies a large elasticity of training with respect to the wage. Additional data from the Apprenticeship Evaluation Survey reveals that the overall effect of a 1% wage increase, including its effect on training, is a 0.1% reduction in a person's self-reported career prospects and a near-zero effect on his/her satisfaction with the apprenticeship.