  We study a totally asymmetric simple exclusion process where jumps happen at
rate one, except at the origin where the rate is lower. We prove a hydrodynamic
scaling limit to a macroscopic profile described by a variational formula. The
limit is valid for all values of the slow rate. The only assumption required is
that a law of large numbers holds for the initial particle distribution. This
includes also deterministic initial configurations. The hydrodynamic
description contains as an unknown parameter the macroscopic rate at the
origin, which is strictly larger than the microscopic slow rate. The limit is
proved by the variational coupling method.
