The answers above are correct, but it might be helpful to see where the factor of $2$ mentioned by @John Doty comes from, and get a sense of scale to get lower pressures (even if not above-Karman-line levels of vacuum).
The altitude $h$ necessary for air pressure to drop by a factor of $e^{-1}$ is approximately given by $PE(h) = \tfrac{1}{2} k_B T$. This comes from comparing the thermal energy with the energy cost to climb a certain height. As the gravitational acceleration close to the Earth's surface is pretty much constant we get $PE = m g h$, solving for $h$ we get
$$
h_{\tfrac{1}{2}} = \frac{\ln(2) k_B T}{2 m g} \approx 6 \text{ km}
$$
as the height needed to half the atmospheric density (pretty close to the real value of $5.5$km I just looked up!). If you want to have only ~$1$% of the pressure, you need to climb almost 7 times this height.
But, the artificial gravity from a spinning station is not constant. The acceleration goes as: $a_r = a_R \tfrac{r}{R}$ where $R$ is the radius of the whole spinning system and $r$ is your current distance from the hub. To get the potential energy we need to integrate this
$$
\text{PE} = -m \int_R^r a_R \tfrac{r}{R} = \tfrac{1}{2} m a_R \left(R^2-r^2\right)
$$
which evaluated at the hub ($r=0$) gives $\tfrac{1}{2} m a_R R$. Assuming that you want the artificial gravity (and the atmosphere) at the rim to be Earth normal ($a_R = g$) then we can solve to give
$$
R_{\tfrac{1}{2}} = \frac{\ln(2) k_B T}{m g} = 2 h_{\tfrac{1}{2}}\approx 12 \text{ km}
$$
so the values for the radius to get a given rarefication is double what it should be on Earth, as stated by @John Doty.
In other words a station with a radius of $12$km will still have half its atmosphere at the hub. To limit air loss you might want the air to be less than 1/1000 times as dense (still far from a vacuum) where it's open, which means your station needs to be at least $240$km in diameter.
Note that all this ignores the complicated thermodynamics of temperature changes and heat transfers and all of that, but it's enough as a back-of-the-envelope calculation to show why the idea is doomed.
