Replication of eukaryotic genomes is highly stochastic, making it difficult to determine the replication dynamics of individual molecules with existing methods. We report a sequencing method for the measurement of replication fork movement on single molecules by detecting nucleotide analog signal currents on extremely long nanopore traces (D-NAscent). Using this method, we detect 5-bromodeoxyuridine (BrdU) incorporated by Saccharomyces cerevisiae to reveal, at a genomic scale and on single molecules, the DNA sequences replicated during a pulse-labeling period. Under conditions of limiting BrdU concentration, D-NAscent detects the differences in BrdU incorporation frequency across individual molecules to reveal the location of active replication origins, fork direction, termination sites, and fork pausing/stalling events. We used sequencing reads of 20-160 kilobases to generate a whole-genome single-molecule map of DNA replication dynamics and discover a class of low-frequency stochastic origins in budding yeast. The D-NAscent software is available at https://github.com/MBoemo/DNAscent.git .