Completely prime ideals have been studied in the case of associative rings by \textit{V. A. Andrunakievich} and \textit{Yu. M. Ryabukhin} [Dokl. Akad. Nauk SSSR 180, 9-11 (1968); English translation in Sov. Math., Dokl. 9, 565-568 (1968; Zbl 0174.328)] and also by \textit{N. H. McCoy} [Colloq. Math. Soc. János Bolyai 6, 147-152 (1973; Zbl 0262.16034)]. The purpose of this note is to extend these results to zero-symmetric (right) near-rings. Let N be such a near-ring. A proper ideal I of N is called a completely prime ideal iff N/I has no nonzero divisors of zero; a completely semiprime ideal iff N/I has no nonzero nilpotent elements; N is called a completely prime (completely semiprime) near-ring iff (0) is a completely prime (completely semiprime) ideal. Let A be an ideal of N. Then the completely prime radical \({\mathcal C}(A)\) of A is defined by \({\mathcal C}(A):=\cap \{I|\) I completely prime ideal of N and \(A\subseteq I\}\), if \(A\neq N\); \({\mathcal C}(N):={\mathcal C}(0).\) We mention the following results: - Let A be a proper ideal of N. Then A is a completely semiprime ideal iff \(A={\mathcal C}(A)\). A corollary of this theorem is the result: A near-ring N without nilpotent elements is a subdirect sum of near-rings without proper zero divisors [cf. \textit{G. Pilz}, Near-rings (1977; Zbl 0349.16015), Theorem 9.36]. If I is a completely semiprime ideal of N, then \({\mathcal C}(I)=I\) is the intersection of all the minimal completely prime ideals containing I. Furthermore, an element-wise characterization of the radical \({\mathcal C}\) is given. Finally, let \({\mathcal M}\) be the class of all completely prime near-rings. Then \({\mathcal U}{\mathcal M}\), the upper radical determined by \({\mathcal M}\), is exactly the class \({\mathcal C}\), i.e. \({\mathcal U}{\mathcal M}=\{A|\) A a near- ring with \(A={\mathcal C}(A)\}\).