Sex chromosomes play a prominent role in development and evolution and have several characteristic features that distinguish them from autosomes. Across diverse taxa, recombination is typically suppressed at the sex-determining region (SDR) and proportionally elevated in the remainder of the chromosome or pseudoautosomal region (PAR). However, in most model taxa the sex chromosomes are ancient and highly differentiated from autosomes, and thus little is known about recombination dynamics of homomorphic sex chromosomes with incipient sex-determining mechanisms. Here we examine male function (pollen production) and female function (fruit production) in crosses of the dioecious octoploid strawberry Fragaria chiloensis in order to map the small and recently evolved SDR controlling both traits and to examine recombination patterns on the young ZW chromosome. The SDR occurs in a narrow 280kb window, in which the maternal recombination rate is lower than in the orthologous paternal region and the genome-wide average rate, but within the range of autosomal rate variation. In contrast to the SDR, the ZW recombination rate in the PAR is much higher than the rates of the ZZ or autosomal linkage groups, substantially overcompensating for the SDR rate. By extensively sequencing sections of the SDR vicinity in several crosses and unrelated plants, we show that W-specific divergence is elevated within a portion of the SDR and find only a single SNP to be in high linkage disequilibrium with sex, suggesting that any W-specific haplotype protected from recombination is not large. We hypothesize that selection for recombination suppression within the small SDR may be weak, but that fluctuating sex ratios could favor elevated recombination in the PAR to remove deleterious mutation on the W. Thus these results illuminate the recombination dynamics of a nascent sex chromosome with a modestly diverged SDR, which could be typical of other dioecious plants.