The tensile strength of an adhesive joint is predicted for a centre-cracked elastic layer, sandwiched between elastic substrates, and subjected to remote tensile stress. A tensile cohesive plastic zone, of Dugdale type, is placed at each crack tip, and the cohesive zone is characterised by a finite strength and a finite toughness. An analytical theory of the fracture strength is developed (and validated by finite element simulations). The macroscopic strength of the adhesive joint is determined as a function of the relative magnitude of crack length, layer thickness, plastic zone size, specimen width and elastic modulus mismatch between layer and substrates. Fracture maps are constructed to reveal competing regimes of behaviour. The maps span the full range of behaviour from a perfectly brittle response (with no crack tip plasticity) to full plastic collapse. When the sum of crack length and cohesive zone length is less than 0.3 times the layer height, the effect of elastic mismatch between substrate and adhesive layer has only a minor influence upon the macroscopic fracture strength. For this case, the cracked adhesive layer behaves as a centre-crack in an infinite solid made from adhesive, and a transition from toughness control to strength control occurs when the crack length is comparable to that of the cohesive zone length. Alternatively, when the sum of crack length and cohesive zone length exceeds 0.3 times the layer height, the elastic mismatch plays a major role; again there is a transition from toughness control to strength control, but it occurs at a ratio of crack length to layer thickness that depends upon both the elastic mismatch and the ratio of cohesive zone length to layer height. The study also highlights the importance of a structural length scale in the form of layer height times modulus mismatch: this scale is on the order of 1 metre when the layer height equals one millimetre and the elastic modulus of the substrate is one thousand times that of the adhesive layer. The in-plane structural dimensions (including crack length) must exceed this structural dimension in order for a remote K-field to exist within the substrate. Experimental validation of the cohesive zone approach is achieved by measuring the sensitivity of fracture strength to crack length and layer height for a centre-cracked strip made from cellulose acetate layer, sandwiched between aluminium alloy substrates.