Molecular machines that assemble polymers in a programmed sequence are fundamental to life. They are also an achievable goal of nanotechnology. Here, we report synthetic molecular machinery made from DNA that controls and records the formation of covalent bonds. We show that an autonomous cascade of DNA hybridization reactions can create oligomers, from building blocks linked by olefin or peptide bonds, with a sequence defined by a reconfigurable molecular program. The system can also be programmed to achieve combinatorial assembly. The sequence of assembly reactions and thus the structure of each oligomer synthesized is recorded in a DNA molecule, which enables this information to be recovered by PCR amplification followed by DNA sequencing.