Let f be a continuous piecewise monotone map of the interval. If any two periodic orbits of f have different itineraries with respect to the partition of the turning points of f, then f is referred to as "nondegenerate". In this paper we prove that a nondegenerate zero entropy continuous piecewise monotone map f has the Shadowing Property if and only if 1) fdows not have neutral periodic points; 2) for each turning point c of f, either the ω-limit set ω(c,f) of c contains no periodic repellors or every periodic repellor in ω(c,f) is a turning point of f in the orbit of c. As an application of this result, the Shadowing Property for the Feigenbaum map is proven.