The segmentation of abnormal regions on dermoscopic images is an important step for automated computer aided diagnosis (CAD) of skin lesions. Recent methods based on fully convolutional networks (FCN) have been very successful for dermoscopic image segmentation. However, they tend to overfit to the visual characteristics that are present in the dominant non-melanoma studies and therefore, perform poorly on the complex visual characteristics exhibited by melanoma studies, which usually consists of fuzzy boundaries and heterogeneous textures. In this paper, we propose a new method for automated skin lesion segmentation that overcomes these limitations via a novel deep class-specific learning approach which learns the important visual characteristics of the skin lesions of each individual class (melanoma vs non-melanoma) on an individual basis. We also introduce a new probability-based, step-wise integration to combine complementary segmentation results derived from individual class-specific learning models. We achieved an average Dice coefficient of 85.66% on the ISBI 2017 Skin Lesion Challenge (SLC), 91.77% on the ISBI 2016 SLC and 92.10% on the PH2 datasets with corresponding Jaccard indices of 77.73%, 85.92% and 85.90%, respectively, for the same datasets. Our experiments on three well-established public benchmark datasets demonstrate that our method is more effective than other state-of-the-art methods for skin lesion segmentation.