Natural-language processing of historical documents is complicated by the abundance of variant spellings and lack of annotated data. A common approach is to normalize the spelling of historical words to modern forms. We explore the suitability of a deep neural network architecture for this task, particularly a deep bi-LSTM network applied on a character level. Our model compares well to previously established normalization algorithms when evaluated on a diverse set of texts from Early New High German. We show that multi-task learning with additional normalization data can improve our model's performance further.