How can a substance be diluted to nothing?
You correctly point out that the number of molecules in a solution is finite and constant, however the volumetric concentration (that is, how many molecules per litre) changes upon dilution. If, for instance, you take one liter of a 1 mol/L solution of ethanol in water ($\approx{6.02\times{10}^{23}}$ ethanol molecules per liter) and add 9 liters of distilled water, you now have a 1 mole of ethanol in 10 liters, giving a concentration of 0.1 mol/L. If you were to draw out 1 liter of this and count the number of ethanol molecules in that liter, you would find approximately $\approx{6.02\times{10}^{22}}$ - a tenfold reduction. The rest of the ethanol is retained in the other 9 liters of solution.

Equivalently, if you were to measure out 100 mL of 1 mol/L ethanol solution and then make it up to 1 liter by adding distilled water, you would also get a 0.1 mol/L solution. This is the kind of thing taken to absurd extremes in homoeopathy, where tiny quantities of solution are repeatedly diluted, with some shaking. Obviously, they don't literally add billions of liters of water to their solutions, but rather pipette out a tiny amount of solution and discard the remainder, adding water to make up the desired volume. This process is repeated many times.

The finite number of molecules present in a solution leads to the concept of the dilution limit, which is where you have carried out sufficient dilutions that statistically speaking, your chances of finding even a single molecule of the dissolved ingredient become vanishingly small and that by extrapolation, you would have to drink approximately a zillion liters of the preparation to get a single molecule of it.

Where did all those other molecules go? They didn't simply disappear - they were poured down the drain during the dilution process, which curiously is the same place money goes when spent on homoeopathy.

Bonus: homoeopathic plutonium! (courtesy of Theodore Gray's periodic table)