Let \(M\) denote a finite area negatively curved surface isometric to the union of finitely many singular surfaces of revolutions of the form \(y=x^r\), \(r>1\) for \(0\leq x\leq 1\), together with connecting surfaces of bounded negative curvature. A geodesic on \(M\) is complete if it does not hit a cusp, corresponding to the point \(x=0\) on the surfaces of revolution, and is singular if it reaches a cusp in finite time. A unit tangent vector is complete or singular if the corresponding geodesics are complete or singular, respectively. \(SM\) denotes the unit tangent bundle, which is undefined at the finite set of cusp points, and let \(f: SM\to SM\) denote the geodesic flow, which is undefined on the singular unit tangent vectors. Let \(X\subset SM\) be the set of complete unit tangent vectors. This set is a flow invariant and dense Baire set. In this paper, the authors study the dynamics of the geodesic flow for a class of non-complete Riemannian metrics on a negatively curved surface \(M\). They show that the geodesic flow is ergodic and that the union of the closed geodesics is dense in \(M\).