Constants in .NET are kinda weird. When you define a constant, the compiler enforces object-oriented conventions in your source code, but when the compiler creates the assemblies, any assembly that references this constant, including the assembly in which you declare it, gets the call to the constant reference replaced with a reference to the same value stored in the local assembly's manifest (which stores most "literals"). So, constants are effectively stored per-assembly, and are "static" to every use of that constant within the assembly. The problem is, if you change the constant, you must recompile any assembly that references the constant in order to refresh the manifests on the other assemblies.
Personally, in your case, I would define that value as static, and expose it through a property. This will prevent the value simply being embedded, and will instead place it in a static copy of the object that will be referenced from one place by any assembly in the current AppDomain that needs it (which basically means it's the same everywhere in the application, except for some special cases involving app pools or subdomains).
More info: How Constant Are Your Constants?