This article proposes alternative measures of immigrant integration founded in information theory. By considering differences in the heterogeneity of outcomes between immigrants and natives, the proposed measures provide robust and non-parametric estimates of the extent to which cohorts remain defined by their national origin. Integration is furthermore premised on equality in the association between economic characteristics and incomes, so that other factors can begin to shape outcomes for im-migrants and natives alike. Results for successive immigrant cohorts in the post-war era are presented using Census income data for the United States. The speed by which the mark of migration on incomes is fad- ing for different cohorts appears to decline significantly over the analyzed period. The paper also presents evidence on the disadvantageous relationship between immigrant status, education and incomes, as well as the integration of immigrants into the US racial and gender hierarchy. Integration appears to be increasingly reserved for college-educated and white men, whereas incomes for Hispanic and less educated immigrants remain strongly shaped by immigrant status.