As magnetization and semiconductor based storage technologies approach their limits, bio-molecules, such as DNA, have been identified as promising media for future storage systems, due to their high storage density (petabytes/gram) and long-term durability (thousands of years). Furthermore, nanopore DNA sequencing enables high-throughput sequencing using devices as small as a USB thumb drive and thus is ideally suited for DNA storage applications. Due to the high insertion/deletion error rates associated with base-called nanopore reads, current approaches rely heavily on consensus among multiple reads and thus incur very high reading costs. We propose a novel approach which overcomes the high error rates in basecalled sequences by integrating a Viterbi error correction decoder with the basecaller, enabling the decoder to exploit the soft information available in the deep learning based basecaller pipeline. Using convolutional codes for error correction, we experimentally observed 3x lower reading costs than the state-of-the-art techniques at comparable writing costs.The code, data and Supplementary Material is available at https://github.com/shubhamchandak94/nanopore_dna_storage.