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.

Here’s an article on Rosemary Propagation from botany.

From the article, “PROPAGATION: Cuttings can be made from firm side shoots, about 9 inches long, in late summer or early fall. In mild climates, they can be inserted in a trench that’s 5 inches deep, outside or in a cold frame. The following autumn, the rooted plants are set out, 1 foot apart, in a nursery bed. In the ensuing year, they may be planted in their permanent places. The tips of the shoots should be pinched often while they are still small to encourage bushiness. Cuttings, 3 to 4 inches long, will form roots quickly, in sand, in a cool greenhouse or terrarium.”

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!