Text-only adaptation of an end-to-end (E2E) model remains a challenging task for automatic speech recognition (ASR). Language model (LM) fusion-based approaches require an additional external LM during inference, significantly increasing the computation cost. To overcome this, we propose an internal LM adaptation (ILMA) of the E2E model using text-only data. Trained with audio-transcript pairs, an E2E model implicitly learns an internal LM that characterizes the token sequence probability which is approximated by the E2E model output after zeroing out the encoder contribution. During ILMA, we fine-tune the internal LM, i.e., the E2E components excluding the encoder, to minimize a cross-entropy loss. To make ILMA effective, it is essential to train the E2E model with an internal LM loss besides the standard E2E loss. Furthermore, we propose to regularize ILMA by minimizing the Kullback-Leibler divergence between the output distributions of the adapted and unadapted internal LMs. ILMA is the most effective when we update only the last linear layer of the joint network. ILMA enables a fast text-only adaptation of the E2E model without increasing the run-time computational cost. Experimented with 30K-hour trained transformer transducer models, ILMA achieves up to 34.9% relative word error rate reduction from the unadapted baseline.