Automatic syllable count estimation (SCE) is used in a variety of applications ranging from speaking rate estimation to detecting social activity from wearable microphones or developmental research concerned with quantifying speech heard by language-learning children in different environments. The majority of previously utilized SCE methods have relied on heuristic DSP methods, and only a small number of bi-directional long short-term memory (BLSTM) approaches have made use of modern machine learning approaches in the SCE task. This paper presents a novel end-to-end method called SylNet for automatic syllable counting from speech, built on the basis of a recent developments in neural network architectures. We describe how the entire model can be optimized directly to minimize SCE error on the training data without annotations aligned at the syllable level, and how it can be adapted to new languages using limited speech data with known syllable counts. Experiments on several different languages reveal that SylNet generalizes to languages beyond its training data and further improves with adaptation. It also outperforms several previously proposed methods for syllabification, including end-to-end BLSTMs.