Chromosomal instability is a common characteristic of many cancers. Chromosomally instable tumour cells exhibit frequent copy number aberrations (CNAs) and a wide variation in the amount of DNA in cancer cells, referred to as cell ploidy. High levels of ploidy, in particular, are associated with whole genome doubling (WGD), a widespread macro-evolutionary event in tumour history. Individual cells' genomes are also undergoing replication as part of the cell cycle, and this constitutes an important covariate for single-cell genome analysis. Accurate and unbiased measurement of single-cell ploidy and replication status, including WGDs, based on DNA sequencing data is important for many downstream applications, such as detecting genomic variants, quantifying intratumour heterogeneity, and reconstructing tumour evolutionary phylogenies. Here we present scAbsolute, an approach to measure ploidy and replication status in single cells using scalable stochastic variational inference with a constrained Dirichlet Process Gaussian Mixture Model. We demonstrate its accuracy across three sequencing technologies (10X, DLP, ACT) and different cell lines and tumour samples. We address the problem of identifying cells with double the amount of DNA, but otherwise identical copy number profiles as is the case after WGD, solely based on sequencing information. Finally, we provide a robust and general method for identifying cells undergoing DNA replication. scAbsolute provides a scalable and unbiased way of ascertaining single-cell ploidy and replication status, paving the way for accurate detection of CNAs and WGDs in single-cell DNA sequencing data.