What will you do with the water once it has collected at the bottom of the slope?

some other information would be useful. What sort of soil do you have? What are you growing? Just how small/big is the problem - is the ground waterlogged or does it just not drain very well after heavy/prolonged rain? How big is the garden?
You might find that digging in lots of coarse grit helps but if the garden is permanently waterlogged and you have ruled out underground watercourses, springs and leaking water mains, the only way to properly drain the area will be with land drains. A single length of land drain near the bottom of the slope could resolve the problem - the cost probably reflects the cost of breaking into the main drain but you could eliminate that by letting you land drain lead into a bog garden.

I'm not convinced by the waterproof membrane idea, for one thing you would have to bury it so deep that it wouldn't be affected by plant roots and you would have to cover the whole of the slope and you would still have the problem of what are you going to do with the water once you've got it at the bottom of the slope.

The cheaper alternative becomes quite expensive when you realized that you have spent your time and money for nothing.

The cheaper alternative that you describe sounds quite expensive, Mdouglas8! Consider that it involves digging up the entire garden to a depth of 30-70 cm, instead of just a number of trenches.

Before parting with any cash, try to find the cause of the problem. Tradesmen will often exaggerate problems when there could be a simple and inexpensive (or free) solution.

We had a problem with the front garden, which slopes gently down towards the front of our house (we're on a hill). Whenever it rained, water would run down the path like a river, and gather in a deep puddle along the front of the porch, and take several hours to drain away. The lawns either side of the path and close to the house were always waterlogged and never dried out. We called in several people to investigate, and all quoted huge amounts of money to install drainage trenches, etc. so we left it.

When we eventually got round to relandscaping the garden, our landscaper removed the turf and took up the paving slabs which made up the path, and he found, as suspected, that the cause of the problem was nothing more than compacted soil (heavy clay which had never been worked by the previous occupants of the 50-year-old property). We laid gravel over weed-suppressing membrane as planned, and relaid the path with 4" gaps between the slabs to avoid any possible recurrence of the puddle problem. That was almost 2 years ago, and we've had no problems since. We've done lots of planting over those two years, and we can safely say that there are is no waterlogging at all.

Of course, your problem might be more complex, but it's worth checking out before having work done that might not be necessary after all.