Spoofing attacks posed by generating artificial speech can severely degrade the performance of a speaker verification system. Recently, many anti-spoofing countermeasures have been proposed for detecting varying types of attacks from synthetic speech to replay presentations. While there are numerous effective defenses reported on standalone anti-spoofing solutions, the integration for speaker verification and spoofing detection systems has obvious benefits. In this paper, we propose a spoofing-robust automatic speaker verification (SR-ASV) system for diverse attacks based on a multi-task learning architecture. This deep learning based model is jointly trained with time-frequency representations from utterances to provide recognition decisions for both tasks simultaneously. Compared with other state-of-the-art systems on the ASVspoof 2017 and 2019 corpora, a substantial improvement of the combined system under different spoofing conditions can be obtained.