Recent syntactic extensions of statistical translation models work with a synchronous context-free or tree-substitution grammar extracted from an automatically parsed parallel corpus. The decoders accompanying these extensions typically exceed quadratic time complexity. This paper extends the Direct Translation Model 2 (DTM2) with syntax while maintaining linear-time decoding. We employ a linear-time parsing algorithm based on an eager, incremental interpretation of Combinatory Categorial Grammar (CCG). As every input word is processed, the local parsing decisions resolve ambiguity eagerly, by selecting a single supertag