In this paper we study distance-regular graphs with intersection array (1){(t+ 1) s, t s,(t− 1)(s+ 1− ψ); 1, 2,(t+ 1) ψ} where s, t, ψ are integers satisfying t≥ 2 and 1≤ ψ≤ s. Geometric distance-regular graphs with diameter three and c 2= 2 have such an intersection array. We first show that if a distance-regular graph with intersection array (1) exists, then s is bounded above by a function in t. Using this we show that for a fixed integer t≥ 2, there are only finitely many distance-regular graphs of order (s, t) with smallest eigenvalue− t− 1, diameter D= 3 and intersection number c 2= 2 except for Hamming graphs with diameter three. Moreover, we will show that if a distance-regular graph with intersection array (1) for t= 2 exists then (s, ψ)=(15, 9). As Gavrilyuk and Makhnev (2013)[9] proved that the case (s, ψ)=(15, 9) does not exist, this enables us to finish the classification of geometric distance-regular graphs with smallest eigenvalue− 3, diameter D≥ 3 and c 2≥ 2 which was started by the first author (Bang, 2013)[1].