For cutting and joining extremely thin metallic wires, the issue of Joule heating in the wires is considered. The middle of a section of a thin Pt wire with a diameter of about 800 nm was melted locally by a direct current supply, and the wire was cut at a predetermined point by applying a force to shear the molten part of the wire. Furthermore, a constant current was applied to a system in which the free ends of two Pt wires were contacted, and the wires were joined together. A parameter, which governs the melting phenomenon at the point of contact of very thin wires, is proposed. It was verified that the conditions required for joining thin wires were able to be determined by the parameter that depends on the applied current, the length and cross sectional area of the wires, and a function of the geometrical quantities for calibrating the thermal conditions.