The presence of low-copy-number regulators and switch-like signal propagation in regulatory networks are expected to increase noise in cellular processes. We developed a noise amplifier that detects fluctuations in the level of low-abundance mRNAs in yeast. The observed fluctuations are not due to the low number of molecules expressed from a gene per se but originate in the random, rare events of gene activation. The frequency of these events and the correlation between stochastic expressions of genes in a single cell depend on the positioning of the genes along the chromosomes. Transcriptional regulators produced by such random expression propagate noise to their target genes.