We consider the following problem: estimate the size of a nonempty set (S\subseteq\left[ N\right] ), given both quantum queries to a membership oracle for (S), and a device that generates equal superpositions (\left\vert S\right\rangle ) over (S) elements. We show that, if (\left\vert S\right\vert ) is neither too large nor too small, then approximate counting with these resources is still quantumly hard. More precisely, any quantum algorithm needs either (Ω\left( \sqrt{N/\left\vert S\right\vert}\right) ) queries or else (Ω\left( \min\left\{ \left\vert S\right\vert ^{1/4},\sqrt{N/\left\vert S\right\vert }\right\} \right)) copies of (\left\vert S\right\rangle ). This means that, in the black-box setting, quantum sampling does not imply approximate counting. The proof uses a novel generalization of the polynomial method of Beals et al. to Laurent polynomials, which can have negative exponents.