The problem is that most rep tubes are made of inferior metal, which tends to weakness and breakage. I would recommend replacing it with a gen or Swiss-made aftermarket tube, which is available from ofrei.com, cousinsuk.com & most other watch supply firms. To remove, you will need to remove the caseback, release the stem (procedure varies depending on the type of movement in your rep - see the Watchmaking section of RWG for general instructions) & use tweezers or a small flathead screw driver to unscrew the broken tube from the inside of the crown. You can use a small, 3-sided file slid into the remaining portion of the tube in the case to unscrew it.
To install the new tube, use the same file to install the new tube in the case. Snug it with moderate finger pressure only; do not force or tighten it as it will snap off! Then, reassemble. If unsure or you lack the tools, by all means, take the watch to a local watchmaker & have him handle the replacement for you. It is a quick & easy job for a professional watchmaker.