Immigrants across Europe earn less and work in lower-quality jobs than natives, but the mechanisms underlying these disparities remain poorly understood. This paper asks whether immigrant disadvantage reflects barriers to accessing good jobs or skill deficits that persist even within similar positions. Using PIAAC Cycle 2 data (2018-2023) for eight European countries, we compare immigrants and natives working in the same occupation-industry cells and performing the same types of tasks. We find that immigrants score 35 to 40 points lower in literacy and numeracy than natives overall, with 70 to 75 percent of this gap persisting within jobs. Immigrants also perform fewer cognitively demanding tasks than natives in similar jobs. However, these task differences disappear entirely once we account for within-job skill gaps, while manual task use shows no immigrant-native differences at all. The evidence points to a skill-mediated mechanism: immigrants perform fewer complex tasks because they have lower cognitive proficiency, not because employers restrict their access to such work. This finding redirects policy attention from workplace discrimination toward skill development and credential recognition as the key margins for improving immigrant labour market integration.