The Spanish local cultivar ‘Ambrunés’ stands out due to its high organoleptic quality and fruit firmness. These characteristics make it an important parent for breeding cherries with excellent fresh and post-harvest quality. In this work, an F₁ sweet cherry population (n = 140) from ‘Ambrunés’ × ‘Sweetheart’ was phenotyped for 2 years for fruit diameter, weight and firmness and genotyped with the RosBREED cherry Illumina Infinium® 6K SNP array v1. These data were used to construct a linkage map and to carry out quantitative trait locus (QTL) mapping of these fruit quality traits. Genotyping of the parental cultivars revealed that ‘Ambrunés’ is highly heterozygous, and its genetic map is the longest reported in the species using the same SNP array. Phenotypic data analyses confirmed a high heritability of fruit size and firmness and a distorted segregation towards softer and smaller fruits. However, individuals with larger and firmer fruits than the parental cultivars were observed, revealing the presence of alleles of breeding interest. In contrast to other genetic backgrounds in which a negative correlation was observed between firmness and size, in this work, no correlation or low positive correlation was detected between both traits. Firmness, diameter and weight QTLs detected validated QTLs previously found for the same traits in the species, and major QTLs for the three traits were located on a narrow region of LG1 of ‘Ambrunés’. Haplotype analyses of these QTLs revealed haplotypes of breeding interest in coupling phase in ‘Ambrunés’, which can be used for the selection of progeny with larger and firmer fruits.