Texas Succession (not secession)

Yesterday, Lieutenant Governor David Dewhurst threw his hat in the ring for Senator Kay Bailey Hutchison’s soon to be vacated seat.

The media keeps telling us that Governor Perry will soon declare his candidacy for President.

So, if Perry and Dewhurst win their (speculative) primaries, then in November, how would the New Governor be chosen? Would Dewhurst resign and let us appoint Abbott, or would we wait ’til the middle of November to start the succession, would we need a special election, or what?