Let R be a commutative ring with nonzero identity. Let J(R) be the set of all ideals of R and let delta : J(R) - -> J(R) be a function. Then delta is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J subset of I, we have L subset of delta(L) and delta(J) subset of delta(I). Let delta be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of delta-primary ideals. A proper ideal I of R is said to be a 1-absorbing delta-primary ideal if whenever nonunit elements a, b, c is an element of R and abc is an element of I, then ab is an element of I or c is an element of delta(I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing delta-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.