Stored procedure to write in a file
As discussed in comments with @scsimon and @Milney, I've created the following table:

CREATE TABLE [dbo].[file_write] 
(
  [ID] bigint NOT NULL IDENTITY(1,1) ,
  [PATH_FILE] nvarchar(500) NOT NULL ,
  [TEXT_FILE] nvarchar(500) NOT NULL ,
  [DATE_WRITE] datetime NOT NULL ,
  PRIMARY KEY ([ID])
)


After that, i've written a stored procedure in order to insert into this table using only PATH_FILE and TEXT_FILE as parameters. This procedure replaces the one I had written in my question. 

CREATE PROCEDURE [dbo].[insertFileWrite]
(
  @PATH_FILE nvarchar(500),
  @TEXT_FILE nvarchar(500)
)
AS
BEGIN
  DECLARE @NOW datetime = CURRENT_TIMESTAMP

  INSERT INTO dbo.file_write(PATH_FILE, TEXT_FILE, DATE_WRITE) 
         VALUES (@PATH_FILE , @TEXT_FILE , @NOW)
END


Finally, i've written a second stored procedure in order to write the lines that are in my table. This stored procedure is executed every 5 minutes by a scheduled task on the server. 

CREATE PROCEDURE [dbo].[writeAllIntoFile]
AS
BEGIN
  DECLARE @PATH_FILE nvarchar(500)
  DECLARE @PATH_FILE_PREV nvarchar(500)
  DECLARE @TEXT_FILE nvarchar(500)
  DECLARE @ID bigint

  DECLARE @OLE int 
  DECLARE @FileID int

  BEGIN TRANSACTION t_writeAllIntoFile
  BEGIN TRY

    DECLARE c_writeFile CURSOR FOR
      SELECT PATH_FILE,
             TEXT_FILE,
             ID
      FROM dbo.file_write
      ORDER BY PATH_FILE ASC,
               DATE_WRITE ASC

    OPEN c_writeFile
    FETCH NEXT FROM c_writeFile INTO @PATH_FILE, @TEXT_FILE, @ID

    SET @PATH_FILE_PREV = @PATH_FILE

    EXECUTE sp_OACreate 'Scripting.FileSystemObject', @OLE OUT 
    EXECUTE sp_OAMethod @OLE, 'OpenTextFile', @FileID OUT, @PATH_FILE, 8, 1 

    WHILE @@FETCH_STATUS = 0
    BEGIN
      IF (@PATH_FILE &lt;&gt; @PATH_FILE_PREV)
      BEGIN
        EXECUTE sp_OADestroy @FileID
        EXECUTE sp_OADestroy @OLE 

        EXECUTE sp_OACreate 'Scripting.FileSystemObject', @OLE OUT 
        EXECUTE sp_OAMethod @OLE, 'OpenTextFile', @FileID OUT, @PATH_FILE, 8, 1 
      END

      EXECUTE sp_OAMethod @FileID, 'WriteLine', NULL, @TEXT_FILE
      DELETE FROM dbo.file_write WHERE ID = @ID

      SET @PATH_FILE_PREV = @PATH_FILE
      FETCH NEXT FROM c_writeFile INTO @PATH_FILE, @TEXT_FILE, @ID
    END

    CLOSE c_writeFile
    DEALLOCATE c_writeFile

    EXECUTE sp_OADestroy @FileID
    EXECUTE sp_OADestroy @OLE 

    COMMIT TRANSACTION t_writeAllIntoFile
  END TRY
  BEGIN CATCH
    ROLLBACK TRANSACTION t_writeAllIntoFile

    IF (SELECT CURSOR_STATUS('global','c_writeFile')) &gt;= -1
    BEGIN
      IF (SELECT CURSOR_STATUS('global','c_writeFile')) &gt; -1
      BEGIN
        CLOSE c_writeFile
      END

      DEALLOCATE c_writeFile
    END

    RETURN -999
  END CATCH
END


Whis this solution, I don't have to open and close my file everytime I want to write one line into it. Since writing is now running on its side, performances have been a lot improved.