We demonstrate operation of a constant-pressure flowmeter capable of generating and accurately measuring flows as low as 2 × 10-13 mol/s. Generation of such small flows is accomplished by using a small conductance element with C ≈ 50 nL/s. Accurate measurement then requires both low outgassing materials (< 1 × 10-15 mol/s) and small volume changes (≈ 70 μL). We outline the present flowmeter's construction, detail its operation, and quantify its uncertainty. The type-B uncertainty is < 0.2 % (k = 1) over the entire operating range. In particular, we present an analysis of its hydraulic system, and quantify the shift and uncertainty due to the slightly compressible oil. Finally, we compare our flowmeter against a NIST standard flowmeter, and find agreement to within 0.5 % (k = 2).