This paper proposes a novel method to improve the accuracy of the American Sign Language fingerspelling recognition. Video sequences from the training set of the __hicagoFSWild_ dataset are first utilized for training a deep neural network of weakly supervised learning to generate frame labels from a sequence label automatically. The network of weakly supervised learning contains the AlexNet and the LSTM. This trained network generates a collection of frame-labeled images from the training video sequences that have Levenshtein distance between the predicted sequence and the sequence label equal to zero. The negative and positive pairs of all fingerspelling gestures are randomly formed from the collected image set. These pairs are adopted to train the Siamese network of the ResNet-50 and the projection function to produce efficient feature representations. The trained Resnet-50 and the projection function are concatenated with the bidirectional LSTM, a fully connected layer, and a softmax layer to form a deep neural network for the American Sign Language fingerspelling recognition. With the training video sequences, video frames corresponding to the video sequences that have Levenshtein distance between the predicted sequence and the sequence label equal to zero are added to the collected image set. The updated collected image set is used to train the Siamese network. The training process, from training the Siamese network to the update of the collected image set, is iterated until the image recognition performance is not further enhanced. The experimental results from the __hicagoFSWild_ dataset show that the proposed method surpasses the existing works in terms of the character error rate.