Dysarthric speech reconstruction (DSR), which aims to improve the quality of dysarthric speech, remains a challenge, not only because we need to restore the speech to be normal, but also must preserve the speaker's identity. The speaker representation extracted by the speaker encoder (SE) optimized for speaker verification has been explored to control the speaker identity. However, the SE may not be able to fully capture the characteristics of dysarthric speakers that are previously unseen. To address this research problem, we propose a novel multi-task learning strategy, i.e., adversarial speaker adaptation (ASA). The primary task of ASA fine-tunes the SE with the speech of the target dysarthric speaker to effectively capture identity-related information, and the secondary task applies adversarial training to avoid the incorporation of abnormal speaking patterns into the reconstructed speech, by regularizing the distribution of reconstructed speech to be close to that of reference speech with high quality. Experiments show that the proposed approach can achieve enhanced speaker similarity and comparable speech naturalness with a strong baseline approach. Compared with dysarthric speech, the reconstructed speech achieves 22.3% and 31.5% absolute word error rate reduction for speakers with moderate and moderate-severe dysarthria respectively. Our demo page is released here: https://wendison.github.io/ASA-DSR-demo/