Thanks for the question. Evolution is a magnificent theory, but it can be quite subtle. Queries like this challenge one to think even more clearly about the matter.
Several others have gone into the answer in detail. As for me, I think I can actually show you why the situation that you describe would never happen - by talking you through a hypothetical scenario. (Branching, in the above post, is to explain how one species can give rise to two distinct ones. But to answer your question, we don't actually even need to consider it.)
Several of the answers have correctly pointed out that evolution is almost always a gradual process. With that in mind, let us observe a population of (say) some hypothetical polar fox over tens of millenia of slowly increasing global temperatures. Now, it is obvious that adaptions that are good for cold temperatures (e.g. heavy fur, white camouflage, etc.) are not the correct ones for hot temperatures.
Thus, as the temperature slowly rises, those members of the species that happen (by change) to have a mutation that allows them to cope slightly better with the slightly hotter temperature will tend to be better at surviving longer. And surviving longer allows the animal to potentially make more offspring than a cold-adapted one. What is the effect of this? Well, over generations, it is obvious that the population will come to be dominated by warmer-adapted animals, since the colder-adapted ones will be have relatively fewer and fewer offspring.
What we have here is a textbook example of evolution. And it can all happen without there ever needing to be a case where a mother gave birth to a new species in one big jump. A son could easily be ever-so-slightly better adapted to the warmth WHILE STILL BEING ABLE TO MATE WITH A SLIGHTLY COLDER-ADAPTED FEMALE. What's more, he would be able to pass this adaptation on to his offspring, and so the mutation could spread.
Now this change is obviously utterly gradual. Imagine coming in and looking at the fossils a few million years later. You would find a lot of cold-adapted fossils (from the long period when the world was cold), and a lot of warm-adapted fossils (from the long period when the world was warm). You might be tempted to label them as different species on morphological grounds. You would also note, from the few fossils showing a transition between cold- and warm-adapted foxes, that the one seemed to have evolved from the other.
So far so good. Now for the tough question: where does the one species start, and the other end? Recall that the difference between a mother and child was tiny in any given generation. Yet, STACKED UP, when you compare the very start to the very end, much change has taken place. So, in a sense, where you want to say one species ends and another begins in this case is entirely arbitrary, and doesn't threaten evolutionary theory in the slightest.
No doubt the end product couldn't have mated with the starting product, but this is an academic point for this discussion - because it never had to.
Does that help?
(Edit in response to secretsauce's edit: Point taken; we are answering in slightly different ways, and yours may well be a better way of explaining it. Enjoyed reading your answer!)
RESPONSE TO THE LATEST UPDATES.
Firstly re: evolution's gradualism. This has attracted a little controversy, but is accepted by the majority of evolutionists. The short answer is that in a body that already works (e.g. the parent), a large aleration is exponentially more likely to kill the organism. Small alterations, on the contrary, while still more likely to do harm than good, at least have better odds. So, most evolutionary change is likely to have been accomplished in small steps. (Large step mutations generally killed off the animal.)
Re: Evidence of gradual speciation. I'm not entirely sure I'm understanding your objection to the second-last update. If I get you, you agree that if evolution were gradual enough, a new mutant would still be able to mate viably with other members of its species (most 'conventional' animals don't mate with their parents!). You just object that no evidence for this has been seen. If I have been accurate so far, then I can do no better than to point you in the direction of the Salamanders surrounding the Central Valley in California (described beautifully by Dawkins in The Ancestor's Tale).
Though the actual tale is more detailed, you can think of the valley as being shaped roughly in the form of an O. Salamanders first colonised the north side, and then populations moved gradually down the east and west sides. As they did so, they began to evolve. However, because the east and west sides were separated, the two populations evolved SEPARATELY.
At ANY STAGE, however, the salamanders could (and still CAN) breed with the individuals a little north and a little south of them. Yet, at the southern end, where the two groups (whose ancestors went west and east respectively) eventually meet, they look very different.. and CANNOT interbreed.
So one species at the north has become 'two' at the south. At each step southwards, remember, the individuals CAN still mate with their slightly 'less evolved' (cringe) northerly cousins. So gradualism overcomes your objections. Usually the 'less evolved' (cringe again) species die out, and we are just left with two species with a common ancestor. In the special case of the Salamanders above, we are luck enough to have all the intermediates still preserved - just to prove to you that speciation is possible. :)