Immigration to the UK has risen over time. Existing studies of the impact of immigration on the wages of native-born workers in the UK have failed to find any significant effect. This is something of a puzzle since Card and Lemieux, (2001) have shown that changes in the relative supply of educated natives do seem to have measurable effects on the wage structure. This paper offers a resolution of this puzzle – natives and immigrants are imperfect substitutes, so that an increase in immigration reduces the wages of immigrants relative to natives. We show this using a pooled time series of British cross-sectional micro data of observations on male wages and employment from the mid-1970s to the mid-2000s. This lack of substitution also means that there is little discernable effect of increased immigration on the wages of native-born workers, but that the only sizeable effect of increased immigration is on the wages of those immigrants who are already here.