Well that all depends. First off, it depends on the type of wire used. With something like THWN-2 you'd get 150A out of #1, for THWN you'd only get 130 and would need to size up to 1/0.
You might also want to consider sizing up for voltage drop issues. 150A over 260 feet on #1 will be almost 12 volts of drop.

As for the neutral, if you are relatively assured of no more than 115A of unbalanced load, OK.

Is this an existant installation? If it isn't and you're subject to the 2008 NEC, or you have other metalic paths between the buildings (water piping, phone, tv), then you also need to run a grounding conductor (#6 would appear to suffice).

Of course it goes without saying that you need a grounding system for the barn.

It is an existing installation. There are no other metallic paths between the buildings. There is a ground wire. It is either #6 or #4, I cannot remember. It is THWN-2. The bonding screw is not connected in the 100 amp subpanel that I want to upgrade to 150 and it is grounded to a six foot grounding rod at the subpanel. My plan is if the wire can handle it then I will replace the 100 amp panel with a 150 amp panel. Do you think I am good to go?

Most 200A panels will only accept up to a 125 branch circuit breaker.
A 150A "sub-panel" is an odd ball.

"...is grounded to a six foot grounding rod at the subpanel."

One rid is not enough unless you show its resistance is less than 25 ohms to earth.

Since doing the measurement involves driving another rod, and only one additional rod is required anyway, simply adding the second rod is usually the easiest solution.