Over the past few years, there has been a significant rise in the number of fall accidents occurring among elderly individuals, a problem that has been accentuated with to the aging population. Researchers and developers have focused their efforts on investigating and creating various fall detection methods that utilize an accelerometer. However, conventional fall detection methods typically target specific positions where accelerometers are placed. In addition, they suffer from low accuracy which can be attributed to the fact that the classification algorithms commonly employed, such as the support vector machine (SVM) and the random forest (RF), are not specialized in making predictions based on time series data. In this paper, we propose the fall detection method based on a long short-term memory (LSTM) neural network, using an accelerometer. In the proposed method, four kinds of possession positions are set: (i) in hand, (ii) inside a chest pocket, (iii) inside a waist pocket, and (iv) in a bag. The acceleration data collected are classified using the LSTM classifies into one of four classes: (i) standing, (ii) walking, (iii) falling, and (iv) lying down. The results of the multi-class classification are further reclassified into two classes, i.e., fall and non-fall. The experimental results demonstrate that our approach outperforms the conventional methods in terms of fall detection accuracy.