Rare DNA-sequence variants hold important clinical and biological information, but existing detection techniques are expensive, complex, allele-specific, or don't allow for significant multiplexing. Here, we report a temperature-robust polymerase-chain-reaction method, which we term blocker displacement amplification (BDA), that selectively amplifies all sequence variants, including single-nucleotide variants (SNVs), within a roughly 20-nucleotide window by 1,000-fold over wild-type sequences. This allows for easy detection and quantitation of hundreds of potential variants originally at ≤0.1% in allele frequency. BDA is compatible with inexpensive thermocycler instrumentation and employs a rationally designed competitive hybridization reaction to achieve comparable enrichment performance across annealing temperatures ranging from 56 °C to 64 °C. To show the sequence generality of BDA, we demonstrate enrichment of 156 SNVs and the reliable detection of single-digit copies. We also show that the BDA detection of rare driver mutations in cell-free DNA samples extracted from the blood plasma of lung-cancer patients is highly consistent with deep sequencing using molecular lineage tags, with a receiver operator characteristic accuracy of 95%.