Evidence identification, optic disc segmentation and automated glaucoma diagnosis are the most clinically significant tasks for clinicians to assess fundus images. However, delivering the three tasks simultaneously is extremely challenging due to the high variability of fundus structure and lack of datasets with complete annotations. In this paper, we propose an innovative Weakly-Supervised Multi-Task Learning method (WSMTL) for accurate evidence identification, optic disc segmentation and automated glaucoma diagnosis. The WSMTL method only uses weak-label data with binary diagnostic labels (normal/glaucoma) for training, while obtains pixel-level segmentation mask and diagnosis for testing. The WSMTL is constituted by a skip and densely connected CNN to capture multi-scale discriminative representation of fundus structure; a well-designed pyramid integration structure to generate high-resolution evidence map for evidence identification, in which the pixels with higher value represent higher confidence to highlight the abnormalities; a constrained clustering branch for optic disc segmentation; and a fully-connected discriminator for automated glaucoma diagnosis. Experimental results show that our proposed WSMTL effectively and simultaneously delivers evidence identification, optic disc segmentation (89.6% TP Dice), and accurate glaucoma diagnosis (92.4% AUC). This endows our WSMTL a great potential for the effective clinical assessment of glaucoma.