Taking advantage of historical census records that include full first and last names, we apply a new approach to measuring the effect of cultural assimilation on economic success for the children of the last great wave of immigrants to the United States. We created a quantitative index of ethnic distinctiveness of first names and show the consequences of ethnic-sounding names for the occupational achievement of the adult children of European migrants. We find a consistent tendency for the children of Irish, Italian, German, and Polish immigrants with more "American"-sounding names to have higher occupational achievement. About one-third of this effect appears to be due to social class differences in name-giving, and the remaining two-thirds to signaling effects of the names themselves. An exception is found for Russian, predominantly Jewish, immigrants, where we find a positive effect of ethnic naming on occupational achievement. The divergent effects of our new measure of cultural assimilation, sometimes hurting and sometimes helping, lend historical empirical support to more recent theories of the advantages of different paths to assimilation. The effects of first names are robust to controls for the ethnic recognizability of last names, suggesting that immigrants' success depended on being perceived as making an effort to assimilate rather than hiding one's origins.