Had you had it for a while? And if so was it a single well grown plant or could it have been a pot of barely rooted cuttings? It looks a lot like a pot of cuttings. If you only just bought it, had it been kept indoors? And have you put it outside suddenly?

Usually when rosemary dies off it dries up. It doesn't flop.That implies that the growth must have been very soft, a symptom of being grown too fast. I think you need to stop watering it so much and put it in a light, sheltered place. Pebbles at the bottom don't do anything for drainage. The pot needs plenty of holes and the mix needs to be very gritty and free draining.

To add to the sage advice above, I hope your plants are not suffering from root rot only because you moved them from a 4 inch to a 10 inch pot. That's a lot of soil sitting at the bottom of the pot waiting to dry out....

I thought I was giving rosemary advice ;-)

Cute :-)