It is generally believed that a positive proportion of number fields (counted according to increasing absolute value of the discriminant), have class number one. This conjecture, and the numerical data that has led to it, have their origins in Gauss’ Disquisitiones Arithmaticae, where it was conjectured that there are infinitely many quadratic fields with class number one, and that exactly nine of these are complex. While the second half of this statement was settled independently by Heegner [He] and Stark [St1], and was reduced to a finite computation by Baker [Ba], the first half of it remains an important open problem. In fact, it is not known whether there exist infinitely many algebraic number fields with class number one. The modest aim of this note is to suggest another naturally arising family of number fields – namely the family of Hilbert class fields of imaginary quadratic fields – suspected of harboring infinitely many class number one fields, and of showing how this question is related to Gauss’ conjecture. This family is also a natural testing ground for a proposed heuristic “principle,” which in vague form reads: The Hilbert class field of a number field with cyclic class group tends to have class number one. We hope to test further and refine this admittedly imprecise statement in the future. Let K = K be an algebraic number field with class number h. The Hilbert class field of K, i.e. its maximal abelian unramified extension, has degree h over K. A finite extension of K with class number one (if it exists) must contain K as a subfield. For i = 0, 1, 2, . . . , define K to be the Hilbert class field of K. Define an invariant `(K) called the “length of the Hilbert class field tower of K” as follows: `(K) is the smallest nonnegative integer i such that K = K, if such an integer exists, and is ∞ otherwise. The latter possibility is equivalent to the statement that every finite extension of K has class number greater than one. In 1964, Golod and Shafarevich [GS] proved that there exist number fields K with `(K) = ∞. The Heegner-Stark-Baker result may be phrased as follows: there are only nine complex quadratic fields K with `(K) = 0. We conjecture that there are infinitely many complex quadratic fields K with `(K) = 1.