We consider a wave equation with point source terms where λ∈C¹[0,T] is a known function such that λ(0)0, α_k∈, δ(·-x_k) is the Dirac delta function at x_k, 1≤k≤N. We discuss the inverse problem of determining point sources {N,α₁,...,α_N,x₁,...,x_N} or {x₁,...,x_N} from observation data u(η,t), 0<t<T with given η∈(0,1) and T>0. We prove uniqueness and stability in determining point sources in terms of the norm in H¹(0,T) of observations. The uniqueness result requires that η is an irrational number and T≥1, and our stability result needs further a priori (but reasonable) information of unknown {x₁,...,x_N}. Moreover, we establish two schemes for reconstructing {x₁,...,x_N} which are stable against errors in L²(0,T).