Early detection of Alzheimer’s disease (AD) is of great importance in clinical practice and research. The fast intervention leads patients to receive appropriate treatment and care at the earliest stages of the disease. This will further result in maximizing the effectiveness of available medicine and therapies which can potentially slow the disease progression and improve patients’ quality of life. This paper proposes a framework for the detection of Alzheimer’s disease using 2D MRI brain images, employing the LeNet-5 architecture and a custom convolutional neural network (CNN). The proposed methods classify images into four classes, including healthy controls, mild cognitive impairment (MCI), early-stage AD, and advanced-stage AD. The performance of the proposed custom CNN is compared with the LeNet-5 model, and the achieved results are evaluated against the existing literature. In comparison to the results reported in the literature, the proposed models are comparable to state-of-the-art approaches for AD classification using MRI images. Furthermore, our custom CNN model demonstrates superior performance, highlighting the importance of tailoring neural network architectures specifically for AD detection.