Answer
300 molecules per cubic centimeter.
Work Step by Step
At such low pressures, gas molecules are far apart, so we assume that the ideal gas law applies.
$$PV=NkT$$
$$\frac{N}{V}=\frac{P}{kT}=\frac{1\times10^{-12}Pa}{(1.38\times10^{-23}J/K)(273\;K)}\approx\frac{3\times10^8}{m^3}$$
This is the number per cubic meter. We multiply by $10^{-6}\;m^3/cm^3$ to find an answer of 300 molecules per cubic centimeter. Given the rough numbers in the problem, our estimate is only good to one significant figure.
