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In this paper, we explore the possibility of building a unified foundation model that can be adapted to both vision-only and text-only tasks. Starting from BERT and ViT, we design a unified transformer consisting of modality-specific tokenizers, a shared transformer encoder, and task-specific output heads. To efficiently pre-train the proposed model jointly on unpaired images and text, we propose two novel techniques: (i) We employ the separately-trained BERT and ViT models as teachers and apply knowledge distillation to provide additional, accurate supervision signals for the joint training; (ii) We propose a novel gradient masking strategy to balance the parameter updates from the image and text pre-training losses. We evaluate the jointly pre-trained transformer by fine-tuning it on image classification tasks and natural language understanding tasks, respectively. The experiments show that the resultant unified foundation transformer works surprisingly well on both the vision-only and text-only tasks, and the proposed knowledge distillation and gradient masking strategy can effectively lift the performance to approach the level of separately-trained models.