Forecasting sea ice concentration (SIC) and sea ice velocity (SIV) in the Arctic Ocean is of great significance as the Arctic environment has been changed by the recent warming climate. Given that physical sea ice models require high computational costs with complex parameterization, deep learning techniques can effectively replace the physical model and improve the performance of sea ice prediction. This study proposes a novel multi-task fully conventional network architecture named hierarchical information-sharing U-net (HIS-Unet) to predict daily SIC and SIV. Instead of learning SIC and SIV separately at each branch, we allow the SIC and SIV layers to share their information and assist each other's prediction through the weighting attention modules (WAMs). Consequently, our HIS-Unet outperforms other statistical approaches, sea ice physical models, and neural networks without such information-sharing units. The improvement of HIS-Unet is more significant to when and where SIC changes seasonally, which implies that the information sharing between SIC and SIV through WAMs helps learn the dynamic changes of SIC and SIV. The weight values of the WAMs imply that SIC information plays a more critical role in SIV prediction, compared to that of SIV information in SIC prediction, and information sharing is more active in marginal ice zones (e.g., East Greenland and Hudson/Baffin Bays) than in the central Arctic.