Alloying enables engineering of the electronic structure of semiconductors for optoelectronic applications. Due to their similar lattice parameters, the two-dimensional semiconducting transition metal dichalcogenides of the MoWSeS group (MX2 where M= Mo or W and X=S or Se) can be grown as high-quality materials with low defect concentrations. Here we investigate the atomic and electronic structure of Mo(1-x)WxS2 alloys using a combination of high-resolution experimental techniques and simulations. Analysis of the Mo and W atomic positions in these alloys, grown by chemical vapour transport, shows that they are randomly distributed, consistent with Monte Carlo simulations that use interaction energies determined from first-principles calculations. Electronic structure parameters are directly determined from angle resolved photoemission spectroscopy measurements. These show that the spin-orbit splitting at the valence band edge increases linearly with W content from MoS2 to WS2, in agreement with linear-scaling density functional theory (LS-DFT) predictions. The spin-orbit splitting at the conduction band edge is predicted to reduce to zero at intermediate compositions. Despite this, polarisation-resolved photoluminescence spectra on monolayer Mo0.5W0.5S2 show significant circular dichroism, indicating that spin-valley locking is retained. These results demonstrate that alloying is an important tool for controlling the electronic structure of MX2 for spintronic and valleytronic applications.