Weak spectral responses in hyperspectral images are often obscured by dominant endmembers and sensor noise, resulting in inaccurate abundance estimation. This paper introduces WS-Net, a deep unmixing framework specifically designed to address weak-signal collapse through state-space modelling and Weak Signal Attention fusion. The network features a multi-resolution wavelet-fused encoder that captures both high-frequency discontinuities and smooth spectral variations with a hybrid backbone that integrates a Mamba state-space branch for efficient long-range dependency modelling. It also incorporates a Weak Signal Attention branch that selectively enhances low-similarity spectral cues. A learnable gating mechanism adaptively fuses both representations, while the decoder leverages KL-divergence-based regularisation to enforce separability between dominant and weak endmembers. Experiments on one simulated and two real datasets (synthetic dataset, Samson, and Apex) demonstrate consistent improvements over six state-of-the-art baselines, achieving up to 55% and 63% reductions in RMSE and SAD, respectively. The framework maintains stable accuracy under low-SNR conditions, particularly for weak endmembers, establishing WS-Net as a robust and computationally efficient benchmark for weak-signal hyperspectral unmixing.