How do I grow a new herb plant from a branch of an existing larger plant?

This seemed like a good, down to earth, guide to it: http://www.nothyme.com/plantpropagation.cfm

The best way to get another plant would be to dig up a small part of your big plant, including the roots. Then repot it. Make sure it is watered well (but don't drown it). Also, if you are putting it into a pot, place some little rocks or pebbles in the bottom of the pot (make sure the pot has drainage hole) and then add the plant and more potting soil.

Use root starter. You can buy it at any nusery. just dip it them plant the cutting and it will grow.

Try putting the sticks of rosemary in dirt in a shady place and keeping them moist. Some plants won’t root in water, just dirt. Root hormone couldn’t hurt either… :) Good luck!