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Let P denote a cubic integral polynomial, and let D(P ) and H(P ) denote the discriminant and height of P respectively. Let N (Q, X) be the number of cubic integer polynomials P such that H(P ) ≤ Q and |D(P )| ≤ X. We obtain the asymptotic formula of N (Q, X) for Q 14/5 ≪ X ≪ Q 4 and as Q → ∞. Using this result, for 0 ≤ η ≤ 0.9 we prove that H(P )≤Q 1≤|D(P )|≪Q 4−η |D(P )| −1/2 ≍ Q 2− η 3 for all sufficiently large Q, where the sum is taken over irreducible polynomials. This improves upon a result of Davenport who dealt with the case η = 0. We also consider an application of the main theorem to some outstanding problems of transcendental number theory.