Abstract <p> We show that the negative-order Korteweg–de Vries equation can be integrated using the inverse spectral problem method. We find the evolution of the spectral data of the Sturm–Liouville operator with a periodic potential associated with the finite-gap solution of the negative-order Korteweg–de Vries equation. The obtained results allow the inverse problem method to be applied to solve the negative-order Korteweg–de Vries equation in the class of periodic functions. We prove important implications regarding the analyticity and the spatial period of the finite-gap solution. We show that the solution constructed by the Dubrovin system of equations and the first trace formula satisfies the negative-order Korteweg–de Vries equation. </p>