Every modern typographer agrees on the one-space rule. It's one of the canonical rules of the profession, in the same way that waiters know that the salad fork goes to the left of the dinner fork and fashion designers know to put men's shirt buttons on the right and women's on the left.
Every major style guide—including the Modern Language Association Style Manual and the Chicago Manual of Style—prescribes a single space after a period. (The Publications Manual of the American Psychological Association, used widely in the social sciences, allows for two spaces in draft manuscripts but recommends one space in published work.)
Most ordinary people would know the one-space rule, too, if it weren't for a quirk of history. In the middle of the last century, a now-outmoded technology—the manual typewriter—invaded the American workplace. To accommodate that machine's shortcomings, everyone began to type wrong. And even though we no longer use typewriters, we all still type like we do. (Also see the persistence of the dreaded Caps Lock key.)
The problem with typewriters was that they used monospaced type—that is, every character occupied an equal amount of horizontal space. This bucked a long tradition of proportional typesetting, in which skinny characters (like I or 1) were given less space than fat ones (like W or M).
Monospaced type gives you text that looks "loose" and uneven; there's a lot of white space between characters and words, so it's more difficult to spot the spaces between sentences immediately. Hence the adoption of the two-space rule—on a typewriter, an extra space after a sentence makes text easier to read. Here's the thing, though: Monospaced fonts went out in the 1970s.
First electric typewriters and then computers began to offer people ways to create text using proportional fonts. Today nearly every font on your PC is proportional. (Courier is the one major exception.) Because we've all switched to modern fonts, adding two spaces after a period no longer enhances readability, typographers say. It diminishes it.