  We present the methodology for ``blind'' millimetre-wave surveys for
redshifted molecular absorption in the CO/HCO$^+$ rotational lines. The
frequency range $30-50$ GHz appears optimal for such surveys, providing
sensitivity to absorbers at $z \gtrsim 0.85$. It is critical that the survey is
``blind'', i.e. based on a radio-selected sample, including sources without
known redshifts. We also report results from the first large survey of this
kind, using the Q-band receiver on the Green Bank Telescope (GBT) to search for
molecular absorption towards 36 sources, 3 without known redshifts, over the
frequency range $39.6 - 49.5$ GHz. The GBT survey has a total redshift path of
$\Delta z \approx 24$, mostly at $0.81 < z < 1.91$, and a sensitivity
sufficient to detect equivalent ${\rm H_2}$ column densities $\gtrsim 3 \times
10^{21}$ cm$^{-2}$ in absorption at $5\sigma$ significance (using the
CO-to-${\rm H_2}$ and HCO$^+$-to-${\rm H_2}$ conversion factors of the Milky
Way). The survey yielded no confirmed detections of molecular absorption,
yielding the $2\sigma$ upper limit $n(z=1.2) < 0.15$ on the redshift number
density of molecular gas at column densities $N({\rm H_2}) \gtrsim 3 \times
10^{21}$ cm$^{-2}$.
