Faulty petrol gauge

Hi, I drive an A6 Quattro (Tiptronic) 2.8 - 2000 model and a little while ago the petrol gauge starting showing incorrect mileage reading. Just to make an example, I'd put in R200.00 worth of petrol today but by morning it will be reading 10kms on the display.

What I would like to know is, is this not telling my car it has no petrol and if so, would this lead to the car stopping in response to this.

How can I fix this problem and what would the cost implications be?

Thanks, Thuli (Durban, Kwa-Zulu Natal, SA)