In this paper we study solvability of equations over free semigroups, known as word equations, particularly Makanin's algorithm, a general procedure to decide if a word equation has a solution. The upper bound time-complexity of Makanin's original decision procedure (1977) was quadruple exponential in the length of the equation, as shown by Jaffar. In 1990 Ko\'scielski and Pacholski reduced it to triple exponential, and conjectured that it could be brought down to double exponential. The present paper proves this conjecture. In fact we prove the stronger fact that its space-complexity is single exponential.