Evolutionary Neural Architecture Search (ENAS) has gained attention for automatically designing neural network architectures. Recent studies use a neural predictor to guide the process, but the high computational costs of gathering training data -- since each label requires fully training an architecture -- make achieving a high-precision predictor with { limited compute budget (i.e., a capped number of fully trained architecture-label pairs)} crucial for ENAS success. This paper introduces ENAS with Dual Contrastive Learning (DCL-ENAS), a novel method that employs two stages of contrastive learning to train the neural predictor. In the first stage, contrastive self-supervised learning is used to learn meaningful representations from neural architectures without requiring labels. In the second stage, fine-tuning with contrastive learning is performed to accurately predict the relative performance of different architectures rather than their absolute performance, which is sufficient to guide the evolutionary search. Across NASBench-101 and NASBench-201, DCL-ENAS achieves the highest validation accuracy, surpassing the strongest published baselines by 0.05\% (ImageNet16-120) to 0.39\% (NASBench-101). On a real-world ECG arrhythmia classification task, DCL-ENAS improves performance by approximately 2.5 percentage points over a manually designed, non-NAS model obtained via random search, while requiring only 7.7 GPU-days.