Abstract: It is known that the category of 2-dimensional topological (quantum) field theories is equivalent to the category of commutative Frobenius algebras. Costello extended this equivalence to cyclic A-infinity algebras by introducing what he called topological conformal field theories, though he did not provide a construction. Kontsevich and Soibelman later sketched a formula for constructing a positive-boundary 2-dimensional topological field theory from a cyclic A-infinity algebra, but they only provided rough ideas. Following their work, we present the explicit construction in full detail. In particular, we describe a procedure for computing the correct signs, which had previously not been discussed.