An automatic-tracking camera apparatus which is capable of realizing continuous and smooth driving and obtaining an image with little position variation of a tracking target from a target position within the image and with little blur. The position of a camera body is changed by a gimbal device. The speed of a tracking target object at the next-after-next start timing of image acquisition by the camera body is predicted. The gimbal device is controlled so that the camera body reaches the position indicated by a position instruction value generated for the next-after-next start timing of image acquisition by the camera body, at the next-after-next start timing, and the speed of the camera body at the next-after-next start timing of image acquisition by the camera body corresponds to the speed predicted for the next-after-next timing of image acquisition.