The article introduces an integration scheme for delay differential equations (DDEs) based on local linearization. The method as introduced is feasible for systems of DDEs with a finite number of fixed delays (even though, it is likely to be extensible to DDEs with variable or distributed delays). It requires knowledge of the Jacobian of the right-hand-side with respect to the current and all delayed arguments. Furthermore, it requires computation of the exponentials of these Jacobians in each step. The authors prove that the scheme is consistent, having a local truncation error of at most order three, and stable, having an overall convergence order of at most two. An example from immunology, a ten-dimensional stiff DDE with up to five different fixed delays demonstrates that the scheme performs well for stiff systems. No comparison is made to standard implicit integration schemes such as RADAR [see \textit{N. Guglielmi} and \textit{E. Hairer}, Computing 67, No.~1, 1--12 (2001; Zbl 0986.65069)] with regard to computational effort.