Use double-sided tape to attach a small block of MDF to the end of the first stopper blank, and hold it in place with the live center in the tail stock. Turn it round at the the same time you round off the blank, and allow the tail-stock live center to embed itself in the MDF. Turn the sides of the stopper, and then remove the MDF to finish the end.
Before doing the next stopper, glue a small bit of sandpaper to the smooth face of the MDF 'puck' (the face opposite the indentation made by the tail stock live center). For subsequent stoppers, simply place the MDF against the end of the blank, and hold it in place with the tailstock. The sandpaper will keep the MDF from spinning against the blank.