Basic dimmer with potentiometer just going on and off

I am trying to make a very simple dimmer for an LED.

1 x Potentiometer
1 x Blue LED
1 x 220ohm resistor
Arduino Uno
Breadboard

I'm powering the breadboard power rails with 5V and GND.
This powers the potentiometer, and a signal wire to A0
The LED is connected with the resistor, to D13

Code:

int DIMMER_PIN = A0;
int LED_PIN = 13;

void setup() {
Serial.begin(9600);
pinMode(DIMMER_PIN, INPUT);
pinMode(LED_PIN, OUTPUT);
}

void loop() {
int Val = analogRead(DIMMER_PIN);
int Brightness = map(Val, 1, 1024, 1, 255);
analogWrite(LED_PIN, Brightness);
Serial.println(String(Val) + " = " + String(Brightness));
}

However, what I am finding is that while the potentiometer is reporting less than 128, the LED is off. When the value from the potentiometer is between 126 and 255 - LED is on.

Example logged output:
22:25:16.508 -> 635 = 158
22:25:16.543 -> 639 = 159
22:25:16.543 -> 643 = 160
22:25:16.543 -> 648 = 161
22:25:16.576 -> 653 = 162
22:25:16.576 -> 657 = 163
22:25:16.576 -> 660 = 164
22:25:16.612 -> 663 = 165
22:25:16.612 -> 666 = 166
22:25:16.646 -> 668 = 166
22:25:16.646 -> 670 = 167

Which seems right. Eg. at 670 on the potentiometer, I'm sending 167 as the brightness value from the AnalogWrite.

Can anyone see why I am not getting a dimmer effect, but instead, and on/off?

Note, I've tried the LED on A5 as well, with the same results.

Pin 13 isn't a PWM capable pin on most Arduino's :wink:

Thanks septillion - Noticed that as I posted.
Need to use a PWM pin.
Fixed.
Thanks.