Because colonizations and invasions are often associated with genetic bottlenecks, they offer an opportunity to directly observe de novo mutations and their subsequent fate. North America has recently been colonized by Arabidopsis thaliana, and many of the individuals found today belong to a single lineage, HPG1. To determine substitution rates under natural conditions in this lineage, we have sequenced 100 HPG1 genomes from plants collected between 1863 and 2006. We infer that the last common HPG1 ancestor lived in the early 17th century, most likely the time when HPG1 began to colonize N. America. Demographic reconstructions infer substantial population size fluctuations during the past four centuries. Even though changing demographics can undermine the effect of natural selection, we observed that mutations at coding sites were at lower frequency than mutations at other sites, consistent with the effect of purifying selection. Exceptionally, some mutations rose to high frequency and some had measurable effects in root development, consistent with positive selection acting over mutations with an adaptive value. Our work showcases how by applying genomics methods to a combination of modern and historic samples we can learn about plant colonisations and invasions and observe evolution in action.