In response to Wnt stimulation, beta-catenin accumulates and activates target genes. Using modeling and experimental analysis, we found that the level of beta-catenin is sensitive to perturbations in the pathway, such that cellular variation would be expected to alter the signaling outcome. One unusual parameter was robust: the fold-change in beta-catenin level (post-Wnt/pre-Wnt). In Xenopus, dorsal-anterior development and target gene expression are robust to perturbations that alter the final level but leave the fold-change intact. These suggest, first, that despite cellular noise, the cell responds reliably to Wnt stimulation by maintaining a robust fold-change in beta-catenin. Second, the transcriptional machinery downstream of the Wnt pathway does not simply read the beta-catenin level after Wnt stimulation but computes fold-changes in beta-catenin. Analogous to Weber's Law in sensory physiology, some gene transcription networks must respond to fold-changes in signals, rather than absolute levels, which may buffer stochastic, genetic, and environmental variation.