Translation of eukaryotic mRNAs is often regulated by nucleotides around the start codon. A purine at position -3 and a guanine at position +4 contribute significantly to enhance the translation efficiency. Algorithms to predict the translation initiation site often fail to predict the start site if the sequence context is not present. We have developed a neural network method to predict the initiation site of mRNA sequences that lack the preferred nucleotides at the positions -3 and +4 surrounding the translation initiation site. Neural networks of various architectures comprising different number of hidden layers were designed and tested for various sizes of windows of nucleotides surrounding translation initiation sites. We found that the neural network with two hidden layers showed a sensitivity of 83% and specificity of 73% indicating a vastly improved performance in successfully predicting the translation initiation site of mRNA sequences with weak Kozak context. WeakAUG server is freely available at http://bioinfo.iitk.ac.in/AUGPred/.