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The external language models (LM) integration remains a challenging task for end-to-end (E2E) automatic speech recognition (ASR) which has no clear division between acoustic and language models. In this work, we propose an internal LM estimation (ILME) method to facilitate a more effective integration of the external LM with all pre-existing E2E models with no additional model training, including the most popular recurrent neural network transducer (RNN-T) and attention-based encoder-decoder (AED) models. Trained with audio-transcript pairs, an E2E model implicitly learns an internal LM that characterizes the training data in the source domain. With ILME, the internal LM scores of an E2E model are estimated and subtracted from the log-linear interpolation between the scores of the E2E model and the external LM. The internal LM scores are approximated as the output of an E2E model when eliminating its acoustic components. ILME can alleviate the domain mismatch between training and testing, or improve the multi-domain E2E ASR. Experimented with 30K-hour trained RNN-T and AED models, ILME achieves up to 15.5% and 6.8% relative word error rate reductions from Shallow Fusion on out-of-domain LibriSpeech and in-domain Microsoft production test sets, respectively.