Append text to file

I have a directory full of hundreds of text files.  I need to loop through each of them, writing (appending) their contents to a single file.

This code is overwriting the target file with each new file the StreamReader opens.  How do I append the text to the bottom of the target file instead of overwriting it?

static void Main(string[] args)
        {
            var sourceDirectory = @"C:\target\new\";
            var txtFiles = Directory.EnumerateFiles(sourceDirectory, "*.txt");

            foreach (string currentFile in txtFiles)
            {
                StreamReader exportedfiles = new StreamReader(currentFile, System.Text.Encoding.Default);

                using (FileStream readStream = File.OpenRead(currentFile))

                //Specify path for the combined file that will be created.
                using (FileStream writeStream = File.OpenWrite("C:\\file.txt"))
                {
                    BinaryReader reader = new BinaryReader(readStream);
                    BinaryWriter writer = new BinaryWriter(writeStream);

                    byte[] buffer = new byte[10240]; //Equals a 10mb buffer.
                    int bytesRead;

                    while ((bytesRead = readStream.Read(buffer, 0, 10240)) > 0)
                    {
                        writeStream.Write(buffer, 0, bytesRead);
                    }
                }
            }