How do I copy the content of a dictionary to an new dictionary in C#?

A one-line version of Amal's answer:

var second = first.Keys.ToDictionary(_ => _, _ => first[_]);

using System;
using System.Collections.Generic;

class Program
{
    static void Main(string[] args)
    {
        Dictionary<string, string> first = new Dictionary<string, string>()
        {
            {"1", "One"},
            {"2", "Two"},
            {"3", "Three"},
            {"4", "Four"},
            {"5", "Five"},
            {"6", "Six"},
            {"7", "Seven"},
            {"8", "Eight"},
            {"9", "Nine"},
            {"0", "Zero"}
        };

        Dictionary<string, string> second = new Dictionary<string, string>();
        foreach (string key in first.Keys)
        {
            second.Add(key, first[key]);
        }

        first["1"] = "newone";
        Console.WriteLine(second["1"]);
    }
}

Assuming you mean you want them to be individual objects, and not references to the same object pass the source dictionary into the destination's constructor:

Dictionary<string, string> d = new Dictionary<string, string>();
Dictionary<string, string> d2 = new Dictionary<string, string>(d);

"so that they are not the same object."

Ambiguity abound - if you do actually want them to be references to the same object:

Dictionary<string, string> d = new Dictionary<string, string>();
Dictionary<string, string> d2 = d;

(Changing either d or d2 after the above will affect both)