Let alpha, b, and T be positive numbers, D = (0, infinity), (D) over bar = [0, infinity), and Omega = D x (0, T]. This article studies the first initial-boundary value problem with a concentrated nonlinear source situated at b, u(t) - u(xx) = alpha delta(x - b)f (u(x, t)) in Omega, u(x, 0) = 0 on (D) over bar, u(0, t) = 0 and u(x, t) -> 0 as x -> infinity for 0 < t <= T, where delta (x) is the Dirac delta function and f is a given function such that lim(u -> c)- f (u) = infinity for some positive constant c, and f (u) and its derivatives f'(u) and f" (u) are positive for 0 <= u < c. The problem has a unique continuous solution u before sup {u (x, t) : 0 <= x < infinity} reaches c(-), and u is a strictly increasing function of t in Omega. It is shown that if sup {u (x, t) : 0 <= x < infinity} reaches c(-), then u attains the value c in a finite time only at the point b. A criterion for u to exist globally and a criterion for u to quench in a finite time are given. It is also shown that there exists a critical position b* for the nonlinear source to be placed such that for b <= b*, u exists for 0 <= t < infinity, and for b > b*, u quenches in a finite time. This also implies that u does not quench in infinite time. The formula for computing b* is also derived.