The pick will become the 10th for the Dolphins, who would have 11 had the Dan Wilkinson trade to Denver not been rescinded because of his failure to take a physical for the Broncos.
"Having ten picks not only will allow us to add youth to our football team, but it also will give us a great deal of flexibility over the course of the draft," said Dolphins general manager Randy Mueller.
Jones' deal with St. Louis reportedly is worth $5.9 million over five years, and the size of it made it unlikely from the start the Dolphins would match. Jones had a tremendous first season with Miami in 2005, but was pretty inconsistent last year.
With Jones gone, the only punter on the roster is recent free agent acquisition Ryan Flinn.