Choose a closed \(B\)-orbit \(Y\subseteq X\), which exists because orbit closures in a complete variety contain closed orbits. Replacing \(X\) by \(Y\), assume \(B\) acts transitively. Since \(B\) is solvable, it has a normal connected subgroup \(N\) such that \(B/N\) is either one-dimensional unipotent or one-dimensional toral after passing through a composition series.

By induction on \(\dim B\), the subgroup \(N\) has a fixed point on the complete variety \(Y\) or the action descends to the quotient group \(B/N\). A one-dimensional connected solvable algebraic group acting on a complete variety has a fixed point: for \(\mathbb{G}_a\), orbit closures are rational complete curves with a point at infinity fixed by the action; for \(\mathbb{G}_m\), the closure of a nonclosed orbit has endpoints fixed by \(\mathbb{G}_m\). Hence \(B\) fixes a point.