What is the proper way to rethrow an exception in C#?

You should always use the following syntax to rethrow an exception. Else you'll stomp the stack trace:

throw;

If you print the trace resulting from throw ex, you'll see that it ends on that statement and not at the real source of the exception.

Basically, it should be deemed a criminal offense to use throw ex.

If there is a need to rethrow an exception that comes from somewhere else (AggregateException, TargetInvocationException) or perhaps another thread, you also shouldn't rethrow it directly. Rather there is the ExceptionDispatchInfo that preserves all the necessary information.

try
{
    methodInfo.Invoke(...);
}
catch (System.Reflection.TargetInvocationException e)
{
    System.Runtime.ExceptionServices.ExceptionDispatchInfo.Capture(e.InnerException).Throw();
    throw; // just to inform the compiler that the flow never leaves the block
}

My preferences is to use

try
{
}
catch (Exception ex)
{
     ...
     throw new Exception ("Add more context here", ex)
}

This preserves the original error, but it allows you to add more context, such as an object ID, a connection string, and stuff like that. Often my exception reporting tool will have five chained exceptions to report, each reporting more detail.

If you throw an exception without a variable (the first example) the stack trace will include the original method that threw the exception.

In the second example, the stack trace will be changed to reflect the current method.

Example:

static string ReadAFile(string fileName) {
    string result = string.Empty;
    try {
        result = File.ReadAllLines(fileName);
    } catch(Exception ex) {
        throw;    // This will show ReadAllLines in the stack trace
        throw ex; // This will show ReadAFile in the stack trace
    }

The first preserves the original stack trace of the exception, the second replaces it with the current location.

Therefore the first is by far the better.

The first is better. If you try to debug the second and look at the call stack you won't see where the original exception came from. There are tricks to keep the call-stack intact (try search, it's been answered before) if you really need to rethrow.

I found that if the exception is thrown in the same method that it is caught, the stack trace is not preserved, for what it's worth.

void testExceptionHandling()
{
    try
    {
        throw new ArithmeticException("illegal expression");
    }
    catch (Exception ex)
    {
        throw;
    }
    finally
    {
        System.Diagnostics.Debug.WriteLine("finally called.");
    }
}

It depends. In a debug build, I want to see the original stack trace with as little effort as possible. In that case, "throw;" fits the bill.

In a release build, however, (a) I want to log the error with the original stack trace included, and once that's done, (b) refashion the error handling to make more sense to the user. Here "Throw Exception" makes sense. It's true that rethrowing the error discards the original stack trace, but a non-developer gets nothing out of seeing stack trace information, so it's okay to rethrow the error.

        void TrySuspectMethod()
        {
            try
            {
                SuspectMethod();
            }
#if DEBUG
            catch
            {
                //Don't log error, let developer see
                //original stack trace easily
                throw;
#else
            catch (Exception ex)
            {
                //Log error for developers and then
                //throw a error with a user-oriented message
                throw new Exception(String.Format
                    ("Dear user, sorry but: {0}", ex.Message));
#endif
            }
        }

The way the question is worded, pitting "Throw:" vs. "Throw ex;" makes it a bit of a red herring. The real choice is between "Throw;" and "Throw Exception," where "Throw ex;" is an unlikely special case of "Throw Exception."