Let \(R\) be an associative ring, not necessarily unital. An involution \(*\) on \(R\) is an additive map on \(R\) such that \((a^*)^*=a\) and \((ab)^*=b^*a^*\) for every \(a\), \(b\in R\). The author studies properties of \(*\)-prime rings, that is rings with involution. Recall that an ideal \(I\) of \(R\) is completely prime whenever, for \(a\), \(b\in R\), the inclusion \(ab\in I\) implies \(a\in I\) or \(b\in I\) (or both). Recall also that an ideal \(I\) which is closed under \(*\) is called \(*\)-ideal; such an ideal is \(*\)-prime if it is prime inside the class of all \(*\)-ideals. Analogously one defines the \(*\)-variant of various notions in the case of rings without involution. In the ordinary case, a ring is fully prime whenever each proper ideal is prime. Hence a ring is \(*\)-fully prime if every proper \(*\)-ideal is \(*\)-prime. The author describes the rings with involution where every \(*\)-ideal is completely \(*\)-prime. He also gives an example of a ring which is \(*\)-fully prime but is not fully prime (in the ordinary sense). The author also describes the structure of the fully \(*\)-prime rings satisfying a polynomial identity. Finally the author describes the additive group of \(*\)-fully prime rings.