The equation of motion for a mass that moves under the influence of a central, inverse-square force is formulated and solved as a problem in complex variables. To find the solution, the constancy of angular momentum is first established using complex variables. Next, the complex position coordinate and complex velocity of the particle are assumed to depend on time in an implicit manner only, through the polar angle phi, and angular momentum conservation is then imposed in the complex equation of motion itself, from the start. The general solution follows after performing but a single elementary integral, integer of d phi e[i phi], and the trajectory is shown to be a conic section. Energy conservation is also established using complex variables. The proposed method of solution is rigorous, yet concise and relatively simple to solve, so it could prove useful in a pedagogical context.