Varves - annual sediment layers, common in glacial lakes - are an important source of paleoclimate information. Manually recording their occurrence, typically by visual inspection, can be both time-consuming and prone to error, leading to several attempts in recent years to at least partially computerize the process. However, existing computerized methods of varve detection still require moderate to large amounts of user interaction - they are semi-automated, rather than fully automated. In light of that, this paper is a step towards fully automatic detection of varves. The presented program, DeepVarveNet - a glacial varve detector built on a convolutional neural network, is designed to automatically delineate annual layers in such lacustrine sediment in digital images of photographed sediment cores. To the best of the authors' knowledge, this is the first approach that applies a convolutional neural network to this task. The performance of DeepVarveNet was assessed on a data set comprising images from seven sediment coring sites, of varying sedimentological properties. They represent three northeast U.S. glacial paleolakes, and glacial paleolake Ojibway. Our testing set contained 1415 identified varves, on which DeepVarveNet demonstrated sensitivity at a level of 0.986 and precision equal to 0.834, exceeding that of BMPix and ANFIS, the existing semi-automated varve identifiers.