There's most likely either a failure in the brake servo assistance unit, or an air leak in the vacuum line which connects to it (maybe a perished hose, failed non-return valve or a cracked manifold).

As there are a number of potential causes for this and it is a safety related issue I'd strongly suggest that you either get an experienced amateur car mechanic to take a look or get the car to a garage for correct diagnosis. It is unlikely to be the brake master cylinder though, so if that's what's quoted then ask for an explanation. If you read the link above and this one then you'll have a better understanding of how it all works and less chance of being conned by a dishonest repairer. Subject to parts availability and the precise cause the repair will either take a few minutes and cost peanuts or will take up to an hour in labour charges and cost up to around £100 for parts (longer and more if the manifold is damaged, but as your car runs fine that's highly unlikely to be the cause).

I hope that my reply has been of some assistance to you and that it helps you to quickly fix your problem. Please take a moment to rate my answer.