The problem of end-to-end learning of a communication system using an autoencoder -- consisting of an encoder, channel, and decoder modeled using neural networks -- has recently been shown to be an effective approach. A challenge faced in the practical adoption of this learning approach is that under changing channel conditions (e.g. a wireless link), it requires frequent retraining of the autoencoder in order to maintain a low decoding error rate. Since retraining is both time consuming and requires a large number of samples, it becomes impractical when the channel distribution is changing quickly. We propose to address this problem using a fast and sample-efficient (few-shot) domain adaptation method that does not change the encoder and decoder networks. Different from conventional training-time unsupervised or semi-supervised domain adaptation, here we have a trained autoencoder from a source distribution that we want to adapt (at test time) to a target distribution using only a small labeled dataset, and no unlabeled data. We focus on a generative channel model based on the Gaussian mixture density network (MDN), and propose a regularized, parameter-efficient adaptation of the MDN using a set of affine transformations. The learned affine transformations are then used to design an optimal transformation at the decoder input to compensate for the distribution shift, and effectively present to the decoder inputs close to the source distribution. Experiments on many simulated distribution changes common to the wireless setting, and a real mmWave FPGA testbed demonstrate the effectiveness of our method at adaptation using very few target domain samples~\footnote{Code for our work: \url{https://github.com/jayaram-r/domain-adaptation-autoencoder}}.