So you setup a blank or two with the dowel glued up like you read about and put the dowel end into your Jacobs or drill chuck only to have the dowel snap while truing the blank. It can easily happen even with tailstock support. Frustrating eh? The torque of the rough cuts is simply too much for the dowel. My first project as a turner seven years ago was bottle stoppers and it was not easy. I tried it with the dowel preglued only to have it snap during roughing out. I tried a pin chuck only to have it loosen up constantly, then the pin falls into the sawdust and vanishes. Nothing seemed to work reliably. Then I saw the light and figured it out.
This is what you need to turn bottlestoppers with minimal problems, some steel rod the same diameter as the dowel (any Hardware store) and a four or three jaw scroll chuck. I do not recommend using a Jacobs or drill chuck since these typically are not safe without tailstock support. Additionally, I need the dowel to travel through the body of the chuck and into the spindle. You'll see why later. You also will need an uncut length of dowel from any hardware store, predrilled corks (any woodturning vendor sells them), a small blank of nice colored wood and some glue.
Start by cutting the length of rod to about four inches in length, then grind the end so it has a sharp prong to bite the blank. Drill your blank about an inch deep. Put the rod into your chuck, slide the blank onto the rod and then use a small piece of wood between your live center spur and the blank when you bring up the tailstock. This way you don't get the small hole bit into the top of your blank. A small piece of larger dowel or a preturned plug works fine. Crank the tailstock a bit to get the rod to bite the inside of the blank and rough out the shape. If the blank is end grain along the axis of the lathe, be careful. You can easily split the blank if there is too much pressure. While roughing and shaping I use my parting tool for flattening the blank's bottom, this ensures a nice even seat to the cork. After I get my shape on all but the point touching the wood at the tailstock I remove the blank. I then glue the blank onto the end of the length of dowel. No, I do NOT cut the dowel then glue. Instead I cut the dowel last this way I get the most out of the length of dowel. After it's nice and secure I then slide the dowel into the chuck. Make sure you put the dowel all the way through so that the blank sits flush against the chuck 's jaws. This makes it much more secure and minimizes vibration. I then finish shaping the top of the stopper. Then I move the stopper out of the chuck a bit and retighten so I can sand and finish the entire stopper. Lastly I remove the stopper and the length of dowel from the chuck glue on the cork and cut the dowel flush at the bandsaw. I then use the rest of the dowel for the next stopper. No more broken dowel. No need to buy a special chuck. Happy turning!