To understand how spatiotemporal chaos may modify material transport, we use direct numerical simulations of the three-dimensional Boussinesq equations and of an advection-diffusion equation to study the transport of a passive tracer by the spiral defect chaos state of a convecting fluid. The simulations show that the transport is diffusive and is enhanced by the spatiotemporal chaos. The enhancement in tracer diffusivity follows two regimes. For large Péclet numbers (that is, small molecular diffusivities of the tracer), we find that the enhancement is proportional to the Péclet number. For small Péclet numbers, the enhancement is proportional to the square root of the Péclet number. We explain the presence of these two regimes in terms of how the local transport depends on the local wave numbers of the convection rolls. For large Péclet numbers, we further find that defects cause the tracer diffusivity to be enhanced locally in the direction orthogonal to the local wave vector but suppressed in the direction of the local wave vector.