One cubic mile at sea level. I will assume that that by this, you mean "one cubic mile of air at mean sea level pressure," since if you were to set a one-cubic-mile cube at sea level, the top would be a mile high and have a significantly lower pressure than the bottom (about 15% lower, to be more precise), thus affecting the amount (and thus the mass) in the cube. So since you say "assume STP," that's what we'll do.

Mean sea level pressure = 101,325 Pa (kg/(m*s²))

Standard temperature = 0°C = 273.15 K

Pressure = density * specific gas constant * temperature

We will also assume typical, Earth-atmosphere, dry air, with the specific gas constant of 287.04 m²/(K*s²).

101325 kg/(m*s²) = density * 287.04 m²/(K*s²) * 273.15 K

Solving for density, we get 1.2923 kg/m³.

Now, using the U.S. definition of mile (5280 feet of 12 inches, each of 0.0254 meters; i.e., 1609.3 m), and cubing it, we get a cubic mile of 4.1682 billion m³. Multiply that by our density, and we get 5.3865 billion kg.

Five billion kilograms? Really? Yeah. Well, shave off that 7% or so for altitude differences, but yeah. A volume as big as a cubic mile can hold a lot of gas. Imagine for a moment that same area filled with water (density = 1000 kg/m³). That same cubic mile (encountered frequently in the oceans) has over 4 trillion kg of mass-- 800 times heavier.

So, unless this is a trick question, my final answer is about 5 billion kg. (No, I'm not converting to pounds. Pounds is a unit of weight and is relative. The customary system has no direct unit for mass.)