We construct a cascading brane model of gravity in which the behavior of the gravitational force law interpolates from (n+4)-dimensional to (n+3)-dimensional all the way down to 4-dimensional from longer to shorter length scales. We show that, at the linearized level, this model exhibits the features necessary for degravitation of the cosmological constant. The model is shown to be ghost free with the addition of suitable brane kinetic operators, and we demonstrate this using a number of independent procedures. Consequently this is a consistent infrared (IR) modification of gravity, providing a promising framework for a dynamical, degravitating solution of the cosmological constant problem.