It is now known that the intersection of two Magnus subgroups Mi = <Yi > (1 = i = 2) in a one-relator group is either the free group F on Y1 n Y2 or the free product of F together with an infinite cyclic group (so-called exceptional intersection). Using this, we give conditions under which two embedding theorems for cyclically presented groups can be obtained. This provides a new method for proving such groups infinite. We also give a combinatorial method for checking the presence of exceptional intersections. © 2007 Elsevier Ltd. All rights reserved.