We introduce a new method for Rectilinear Steiner Tree (RST) construction in a graph, using answer set programming. This method provides a formal representation of the problem as a logic program whose answer sets correspond to solutions. The answer sets for a logic program can be computed by special systems called answer set solvers. We describe the method for RST construction in the context of VLSI routing where multiple pins in a given placement of a chip are connected by an RST. Our method is different from the existing methods mainly in three ways. First, it always correctly determines whether a given RST routing problem is solvable, and it always produces a solution if one exists. Second, some enhancements of the basic problem, in which lengths of wires connecting the source pin to sink pins are restricted, can be easily represented by adding some rules. Our method guarantees to find a tree if one exists, even when the total wire length is not minimum. Third, routing problems with the presence of obstacles can be solved. With this approach, we have computed solutions to some RST routing problems using the answer set solver CMODELS.