The fabrication of heterostructures of two-dimensional semiconductors with specific bandgaps is an important approach to realizing the full potential of these materials in electronic and optoelectronic devices. Several groups have recently reported the direct growth of lateral and vertical heterostructures based on monolayers of typical semiconducting transition metal dichalcogenides (TMDCs) such as WSe₂, MoSe₂, WS₂, and MoS₂. Here, we demonstrate the single-step direct growth of lateral and vertical heterostructures based on bandgap-tunable Mo_1-xW_xS₂ alloy monolayers by the sulfurization of patterned thin films of WO₃ and MoO₃. These patterned films are capable of generating a wide variety of concentration gradients by the diffusion of transition metals during the crystal growth phase. Under high temperatures, this leads to the formation of monolayer crystals of Mo_1-xW_xS₂ alloys with various compositions and bandgaps, depending on the positions of the crystals on the substrates. Heterostructures of these alloys are obtained through stepwise changes in the ratio of W/Mo within a single domain during low-temperature growth. The stabilization of the monolayer Mo_1-xW_xS₂ alloys, which often degrade even under gentle conditions, was accomplished by coating the alloys with other monolayers. The present findings demonstrate an efficient means of both studying and optimizing the optical and electrical properties of TMDC-based heterostructures to allow use of the materials in future device applications.