Given the Diophantine equation a2+b2+c2=3abc, a solution triple of natural numbers (a, b, c) can be arranged in ascending order so that a[les ]b[les ]c. Then, given the largest element c, one can ask whether this uniquely determines the triple. This is referred to as the Markoff conjecture. The paper proves that, if c is prime, then there is indeed only one triple that solves the equation with c as the largest element. The proof uses only standard algebraic number theory, but it was prompted by geometric considerations.