A process for welding a first part and a second part, includes: a linear friction step during which the two parts are rubbed against one another in a linear movement, a first cooling step during which the temperature of the parts thus welded is reduced to ambient temperature, a heating step during which the parts thus welded are heated to a temperature above a temperature of transition between a two-phase α-β field and a single-phase β field of the material constituting the two parts, and a second cooling step during which the temperature of the parts thus heated is reduced to ambient temperature. Such a process makes it possible to transform the structure of the welding zone and of the whole part thus produced into a lamellar α structure in a transformed β matrix which exhibits better damage tolerance.