galvanized steel downspout inside wall cavity is wet

Hi,
I have a galvanized steel downspout inside the cavity of the wall in my house. The downspout is between the siding and the inside drywall and runs alongside a stud.
The problem I have is that the downspout is wet. I removed the drywall for a different project and it rained that day. I double checked for leaks but could not find one. Since the water seems to be evenly distributed over the surface of the downspout, I suspect that this is condensation rather than a leak. Also the downspout is cold with rain running down the inside, while it is warm in the house.
My question is, if I cover the downspout back up with insulation and drywall, will it be dry? My idea is that if the downspout is not exposed to the moisture in the air on the inside of the house, it cannot condense the water. My fear is that it will remain damp underneath and grow mold.
Your thoughts?
Thanks, Clarence.

I have never seen a house that did not have the downspouts OUTSIDE the wall, and I would not recommend otherwise.

If I had no other option, I would refit it with PVC pipe and apply lagging to it, before insulating and closing the wall.

Sure does sound like condensation. That metal downspout is going to stay cold whether you stuf it with insulation or not.

Remove the downspout from the wall, insulate the wall, and install an exterior downspout in its place.

You sure it's a downspout and not part of the vent system? If it is a vent, and it's not flashed well through the roof, it could have water flowing down the outside.

For there to be condensation, there would need to be air flow and moisture. In a well constructed wall, you should not have either.