Rassias (2001) introduced the pioneering cubic functional equation in the history of mathematical analysis: f(x+2y)−3f(x+y)+3f(x)−f(x−y)=6f(y) and solved the pertinent famous Ulam stability problem for this inspiring equation. This Rassias cubic functional equation was the historic transition from the following famous Euler-Lagrange-Rassias quadratic functional equation: f(x+y)−2f(x)+f(x−y)=2f(y) to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation: f(x+3y)−3f(x+y)+3f(y−x)−f(x−3y)=48f(y) in fuzzy normed linear spaces. We use the definition of fuzzy normed linear spaces to establish a fuzzy version of a generalized Hyers-Ulam-Rassias stability for above equation in the fuzzy normed linear space setting. The fuzzy sequentially continuity of the cubic mappings is discussed.