Alzheimer's disease (AD) is a neurodegenerative disorder with no known cure that affects tens of millions of people worldwide. Early detection of AD is critical for timely intervention to halt or slow the progression of the disease. In this study, we propose a Transformer model for predicting the stage of AD progression at a subject's next clinical visit using features from a sequence of visits extracted from the subject's visit history. We also rigorously compare our model to recurrent neural networks (RNNs) such as long short-term memory (LSTM), gated recurrent unit (GRU), and minimalRNN and assess their performances based on factors such as the length of prior visits and data imbalance. We test the importance of different feature categories and visit history, as well as compare the model to a newer Transformer-based model optimized for time series. Our model demonstrates strong predictive performance despite missing visits and missing features in available visits, particularly in identifying converter subjects -- individuals transitioning to more severe disease stages -- an area that has posed significant challenges in longitudinal prediction. The results highlight the model's potential in enhancing early diagnosis and patient outcomes.