This paper proposes a modified two-point block backward differentiation formula (MBBDF) for the numerical solution of highly stiff and oscillatory ordinary differential equations. The method achieves third-order accuracy with a reduced error constant and computes two solution values simultaneously at each step. For the selected parameter value $\rho=-\dfrac{7}{8}$, stability analysis shows that the resulting scheme is A-stable, making it suitable for the numerical integration of stiff initial value problems. The method is also consistent and zero-stable, and hence convergent. Numerical experiments on benchmark stiff problems show that the relative accuracy of the considered methods depends o... More >