It's a wonderful old Conn Sousaphone, but not the first Sousaphone, as that was built by J. W. Pepper in 1895. And it's not even the first
Conn Sousaphone, as the one shown in the photo can be dated no earlier than 1903. Conn introduced his first version of the Sousaphone in 1898. But the model Rex is holding, and perhaps that exact horn, was played in Sousa's Band from 1903 until Sousa's death in 1932. That version was played by such giants as Herman Conrad, Jack Richardson, and Bill Bell.
Here's a link to some of my research on the J. W. Pepper Sousaphone, if you're interested:
http://blogs.jwpepper.com/index.php/j-w ... ousaphone/