I have no doubt that the article was genuinely published in The Hampshire Advertiser on the date specified. I haven't been able to find it, but the evidence is strong nonetheless.
It's important to note that newspapers tend to just reprint articles that other newspapers already published. They do this ridiculously often. In fact, that's why it says "Galignani's Messenger" at the bottom of your clipping; that's where The Hampshire Advertiser got it from. So here's a pastiche of the quote from different sources:
South Australian Register, Thu 18 Nov 1852:
The London Times, of July 29, quotes an article
with the above heading from Galignani's Messenger.
It is as follows:
Leicester Chronicle Saturday, July 17, 1852:
The excessive heat which prevails at present gives some interest to the following account of remarkably hot summers: "In 1132 the earth opened, and the rivers and springs disappeared, in Alsace.
Daily Dispatch, 5 August 1852:
The Rhine was dried up. In 1152 the heat was so great that eggs were cooked in the sand.
New Orleans Daily Crescent, September 13, 1852:
In 1160, at the a battle of Bela, a great number of soldiers died from the heat. In 1276 and 1277, in Prance, an absolute failure of the crops of grass and oats occurred.
Lewistown Gazette, September 17, 1852:
In 1303 and 1304, the Seine, the Loire, the Rhine, and the Danube, were passed over dry-footed. In 1303 and 1391, great numbers of animals fell dead, and the crops were scorched up.
The Planters' Banner, September 25, 1852:
In 1440 the heat was excessive. In 1538, '39, '40. '41, the rivers were almost entirely dried up. In 1556 there was a great drought all over Europe. In 1615 and 1616 the heat was overwhelming in France, Italy, and the Netherlands. In 1646 there were 58 consecutive days of excessive heat.
....and so on. There are several more newspapers I found that have the quote and I haven't gotten around to searching in some of the other newspaper databases I have access to.