Thread Function and access

Hello,
I have a written a Thread class (CMyThread) and have a question
concerning the Thread Safety.
The reason for using a Thread is because i need to check the Version of
Application (multilanguage) online by downloading the version string.
Normally this action takes several seconds.

So i decided to call the Run() method of CMyThread, for example:
Run("Connection failed", "New Version available", "You're using the
latest Version" ...);

For easy explanation i will use only one string in the following class.
My question is, if it's safe to access this.m_msg in the Run Method. Or
is there any way to tell the Thread to call a method of my CMyThread
after (!) executiuon to display the version check result?!

Thanks for any answers and best regards

class CMyThread
{
   private string m_msg;

   bool Start(string msg)
   {
      // Save string for later access in Run Method
      this.m_msg = msg;
      _beginthreadex(NULL, 0, &Starter, this, 0, &m_ThreadID);
   }

   // The one and only Thread Function for performing some work
   unsigned Run()
   {
      // Is it safe to access this.m_msg here?!
      // For example:
      MessageBox(NULL, this.m_msg.c_str(), "", NULL);
      // or...
      this.m_msg = "some text";
      // ...
      return 0;
   }

   static unsigned __stdcall Starter(void* pVoid)
   {
      CMyThread *pInst = (CMyThread*)pVoid;
      unsigned iRet = pInst->Run();
      _endthreadex(iRet);
      return iRet;
   }
};