Breadboard arduino + disconnected Serial USB cable == Reset?

Hello,

I have the following scenario:

Everything works perfectly. The serial data can be received when the USB cable is connected, the LED blinks. All is well.

... until I disconnect the USB cable from the Serial USB adapter. Once the external power source is turned on, the sketch runs for a few seconds (I can tell from the blinking LED on pin 9), then resets (LED momentarily stops blinking, LED on pin 13 flickers). Then runs for a few seconds more, then resets -- over and over again (~5 seconds between resets, possibly dependent on the length of serial data I transmit). If I remove the 'auto-reset' capacitor from the board -- this reset does not happen. If I remove the USB serial adapter entirely -- this reset does not happen.

This means I cannot use the USB/Serial adapter permanently in a project, whilst still leaving auto-upload enabled. Doing so would mean that every time the USB cable is disconnected, the Arduino will reset every ~5 seconds.

I would like the USB/Serial adapter to just discard any data when the cable is not connected. I had understood the reset pin on the adapter would be used purely during initial USB connection (and for upload) -- these resets I can live with. I just cannot live with the frequent every-5-seconds reset caused when the USB cable is not connected.

Perhaps I'm misunderstanding how this is supposed to work? All help appreciated.

Thanks.