Let \(V_3\) be the Veronese surface of order 3, and \(S_3\) be the branch curve of a generic projection of \(V_3\) on \(\mathbb{C} P^2\). The authors compute the fundamental group \(\pi (\mathbb{C}^2\smallsetminus S_3)\). They use the technique developed in the preceding parts [Part: I: Contemp. Math. 78, 425-555 (1988; Zbl 0674.14019); Part II: Lect. Notes Math. 1479, 131-180 (1991; Zbl 0764.14014)]; Part III: Contemp. Math. 162, 313-332 (1994; Zbl 0815.14023); Part IV: ibid. 162, 333-358 (1994; Zbl 0815.14024)]. The authors also announce a similar result concerning \(\pi (\mathbb{C} P^2\smallsetminus S_3)\).