It is generally recognized that there is no explicit method that can be A-stable among the linear multi-step methods. However, a novel problem-dependent method is shown to be one-step, explicit, A-stable and second order accurate. To emphasize that the new method can possess the repelling properties of the A-stability and explicit implementation, a variety of first order stiff problems are solved and the results are consistent with the analytical predictions. Since A-stability and second order accuracy allow the use of a large step size and explicit implementation implies no nonlinear iterations, thus it is computationally efficient for solving stiff problems. The trapezoidal method has the same properties except for explicit formulation. It is also adopted in this study for comparison. Numerical tests reveal that the CPU demand for the new method is as small as 1.9%, 1.3%, 1.0% and 0.51% of that consumed by the trapezoidal method for a set of 250, 500, 1000 and 2000 first order ODEs. Consequently, it is computationally much more cheap than the trapezoidal method in the solution of stiff problems.