Serial communication stops when I connect the tx cable
To have two serial ports on Arduino board like Uno use SoftwareSerial.h.

Example

#include &lt;SoftwareSerial.h&gt;

SoftwareSerial mySerial(2,3); // RX, TX

void setup()  
{
  // Open serial communications and wait for port to open:
  Serial.begin(9600);


  Serial.println("Hello!");

  // set the data rate for the SoftwareSerial port
  mySerial.begin(9600);
}

void loop() // run over and over
{
  if (mySerial.available())
    Serial.write(mySerial.read());
  if (Serial.available())
    mySerial.write(Serial.read());
}


Now after opening Serial monitor you should see Hello! and then you can test SoftwareSerial. 

DO NOT FORGET!!!

When you connect to pins 2,3 or etc, this is TTL voltage level. You can't connect direct to RS232.