Elucidating the role of molecular stochasticity in cellular growth is central to understanding phenotypic heterogeneity and the stability of cellular proliferation. The inherent stochasticity of metabolic reaction events should have negligible effect, because of averaging over the many reaction events contributing to growth. Indeed, metabolism and growth are often considered to be constant for fixed conditions. Stochastic fluctuations in the expression level of metabolic enzymes could produce variations in the reactions they catalyse. However, whether such molecular fluctuations can affect growth is unclear, given the various stabilizing regulatory mechanisms, the slow adjustment of key cellular components such as ribosomes, and the secretion and buffering of excess metabolites. Here we use time-lapse microscopy to measure fluctuations in the instantaneous growth rate of single cells of Escherichia coli, and quantify time-resolved cross-correlations with the expression of lac genes and enzymes in central metabolism. We show that expression fluctuations of catabolically active enzymes can propagate and cause growth fluctuations, with transmission depending on the limitation of the enzyme to growth. Conversely, growth fluctuations propagate back to perturb expression. Accordingly, enzymes were found to transmit noise to other unrelated genes via growth. Homeostasis is promoted by a noise-cancelling mechanism that exploits fluctuations in the dilution of proteins by cell-volume expansion. The results indicate that molecular noise is propagated not only by regulatory proteins but also by metabolic reactions. They also suggest that cellular metabolism is inherently stochastic, and a generic source of phenotypic heterogeneity.