Excel: macro to export worksheet as CSV file without leaving my current Excel sheet

There are a lot of questions here to create a macro to save a worksheet as a CSV file. All the answers use the SaveAs, like this one from SuperUser. They basically say to create a VBA function like this:

Sub SaveAsCSV()
    ActiveWorkbook.SaveAs FileFormat:=clCSV, CreateBackup:=False
End Sub

This is a great answer, but I want to do an export instead of Save As. When the SaveAs is executed it causes me two annoyances:

• My current working file becomes a CSV file. I'd like to continue working in my original .xlsm file, but to export the contents of the current worksheet to a CSV file with the same name.
• A dialog appears asking me confirm that I'd like to rewrite the CSV file.

Is it possible to just export the current worksheet as a file, but to continue working in my original file?

Almost what I wanted @Ralph, but here is the best answer, because it solves some annoyances in your code:

1. it exports the current sheet, instead of just the hardcoded sheet named "Sheet1";
2. it exports to a file named as the current sheet
3. it respects the locale separation char.
4. You continue editing your xlsx file, instead of editing the exported CSV.

To solve these problems, and meet all my requirements, I've adapted the code from here. I've cleaned it a little to make it more readable.

Option Explicit
Sub ExportAsCSV()
 
    Dim MyFileName As String
    Dim CurrentWB As Workbook, TempWB As Workbook
     
    Set CurrentWB = ActiveWorkbook
    ActiveWorkbook.ActiveSheet.UsedRange.Copy
 
    Set TempWB = Application.Workbooks.Add(1)
    With TempWB.Sheets(1).Range("A1")
      .PasteSpecial xlPasteValues
      .PasteSpecial xlPasteFormats
    End With        

    Dim Change below to "- 4"  to become compatible with .xls files
    MyFileName = CurrentWB.Path & "\" & Left(CurrentWB.Name, Len(CurrentWB.Name) - 5) & ".csv"
     
    Application.DisplayAlerts = False
    TempWB.SaveAs Filename:=MyFileName, FileFormat:=xlCSV, CreateBackup:=False, Local:=True
    TempWB.Close SaveChanges:=False
    Application.DisplayAlerts = True
End Sub

Note some characteristics of the code above:

1. It works just if the current filename has 4 letters, like .xlsm. Wouldn't work in .xls excel old files. For file extensions of 3 chars, you must change the - 5 to - 4 when setting MyFileName in the code above.
2. As a collateral effect, your clipboard will be substituted with current sheet contents.

Edit: put Local:=True to save with my locale CSV delimiter.