Why can’t I have “public static const string S = “stuff”; in my Class?

When trying to compile my class I get an error:

The constant 'NamespaceName.ClassName.CONST_NAME' cannot be marked static.

at the line:

public static const string CONST_NAME = "blah";

I could do this all of the time in Java. What am I doing wrong? And why doesn’t it let me do this?