Am I required to reveal my employment status to my soon to be ex employer. See details?

I work in a hospital but through an out of state contractor. So actually not directly for the hospital. The contractor will no longer be my employee after August 31. I was told 3 months ago that I will receive a two week severance. There were no strings attached when I was told that. Now I am told I will only receive the severance if I do not have a job.
The hospital has hired me beginning September 1. Same job, same pay, etc.
Am I required to tell my old company that I have a job?