I agree that Laos is one syllable. It rhymes with “house” and is not pronounced “lay-ose” like I’ve heard some people say. The Lao people themselves don’t even include the S; to them, it’s Lao, rhymes with “cow.”
Siam (ok Sam, I know it’s been renamed).
For the people posting Éire, how are you pronouncing it that it’s one syllable? It’s two in Irish, and this dictionary lists four two-syllable pronunciations but no one-syllable ones.
And if Wales isn’t a country, England can’t be a country either.
Siam is two syllables.
Without looking:
Spain
France
Wales
Chad
I think they may have considered Wales as a part of the UK (like Nebraska is part of the US and Alberta is part of Canada and Queensland is a part of Australia).
Please note I’m just trying to guess the reasons the trivia book people didn’t include Wales–I’m not advocating it’s a good or valid reason to not include it. : p
Haha! A Brit of my acquaintance visibly restrained himself from ripping me a new asshole once when I facetiously made a similar comparison. He did let me know in no uncertain terms that Wales, Scotland etc were all separate countries banded together under the UK.
Well, it’s a good thing I’m not asserting that myself then…that I’m just guessing why they didn’t count it. 
In English, or in the name of the country in its own national language(s)? I think the latter would be more interesting.
In Thai, Thailand is called Prathet Thai, which literally means “Thai country,” so Prathet or “Country” is more of an identifier. (They tack Prathet onto the beginning of every country. Prathet Farangset is “France Country.”) Muang Thai, or “Thai Homeland,” is also common. In both cases, the actual name would be just “Thai.”
I came up with Chad, Spain, France & Wales, then quit Greece-less because I won.
mmm
My guess without checking anything:
France
Wales (I bet this doesn’t count, although it should, and I’ve missed one)
Spain
Chad
Took me approx two and half minutes.
Me too.
In my dialect (confirmed by Wikipedia’s pronunciation recording) Wales has two distinct vowel sounds. ‘Way-ulls’. How do you pronounce it so it’s only one?
(Of course we call it ‘Lay-ohs’ too, but I knew better and that was one of the four I came up with. I missed Greece.)
Wales is not a country - it is a principality.
There was a union of the crowns of England, Scotland and Ireland (of which Northern Ireland is the remenant) but not Wales. It was conquered.
Now, who can name all the countries on Earth with a name that ends in -gry?
It doesn’t because it’s two syllables.
“Principality” and “country” are not mutually exclusive; Monaco and Liechtenstein are principalities.
Regarding the union of the crowns, Poland was partitioned out of existence altogether in the past, and Croatia was subsumed into Yugoslavia; does that mean that Poland and Croatia are now not countries? Wales was conquered in 1282, but it wasn’t merged with England until 1536, so conquered =/= merged. Even that merger was clearly more show than substance: nearly 500 years later, Wales has its own language, history, culture, laws, legislative body (the Welsh Assembly), international representation (e.g. the Wales team in World Cup soccer or Six Nations rugby).
It’s not an independent nation-state, but I don’t think the 1282 conquest or Charles’s ceremonial title “Prince of Wales” are proof that Wales is not a country.
It probably would have been clearer if the OP had specified either UN members, or countries that are widely recognized as sovereign, either of which would exclude Wales. By either of these criteria, there are only four countries that have one-syllable names in English (France, Spain, Greece, Chad); or five if the one-syllable pronunciation of Laos is accepted.
I don’t think that most people would consider Wales to be any more of a country than Catalonia or Hawaii are.
One syllable US City Names is also rare. Unless I misread the list, this Wiki link has
Flint Michigan
has the largest city with a total of one syllable. and it is 269th. Offhand, I can only think of York Pa, Rome Ga (and NY) with one syllable.