The immersed boundary (IB) method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures. It is characterized by the use of a uniform Cartesian mesh for the fluid, a Lagrangian curvilinear mesh on the elastic material, and discrete delta functions for communication between the two grids. We consider a simple IB problem in a two-dimensional periodic fluid domain with a one-dimensional force generator. We obtain error estimates for the velocity field of the IB solution for the stationary Stokes problem. We use this result to prove convergence of a simple small-amplitude dynamic problem. We test our error estimates against computational experiments.