With its line of Chromebooks made by various hardware partners, Google (GOOG) began testing in earnest the idea that tech users could live their digital lives almost entirely in the cloud. The low-cost laptops, which are powered by the Chrome operating system first went on sale in 2011. They only run Web-based applications, not traditional desktop programs, and eschew the trappings of traditional laptops such as an optical disc reader or large hard drive.

After a sluggish start, Chromebooks are gaining momentum. But by and large, people buy them because they are cheap. I personally raved about the $250 Samsung Chromebook not because it’s a great computer, but because a) I already have a couple of Macs around the house, and b) at that price, I could afford to get an extra laptop and put up with the shortcomings of a Web-only device: no iTunes or iPhoto, no Skype, no Minecraft (if you don’t know what that is, ask any third-grade boy), and subpar versions of word processing and spreadsheet programs.

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Now Google is making a bet that some people will want to live entirely in the cloud — not primarily because it is cheaper — but because it is better. On Thursday, it unveiled the Chromebook Pixel, a laptop as stylish and fast as any ultrabook. (That category was created by Intel (INTC) to help boost flagging PC sales; so far, no luck.) Google’s Pixel has a full-sized keyboard and a super-high-resolution display that works as a touchscreen, like many newer PCs running Microsoft (MSFT) Windows 8.

“We think this is a real game changer in how people can start living the cloud,” said Sundar Pichai, the senior vice president in charge of Chrome and apps at Google, as he unveiled the Pixel in San Francisco on Thursday. Maybe. But the catch is that the Pixel, which will ship next week, costs a jaw-dropping $1,299 for a WiFi-only model. A version with LTE to connect to high-speed cellular networks is coming in April and will cost $1,449.

That’s likely to be a tough sell. For $100 less than the base model, you can get a 13-inch MacBook Air from Apple (AAPL), the popularity of which essentially spurred the creation of the ultrabook category in the first place. The Air may not have a touch screen, but you can do all the cloud computing you want and run just about any desktop program. You can store your photos, music and videos, and yes, back them up in the cloud if you want.

So who will buy the Pixel? Pichai says that plenty of businesses have already embraced the concept of cloud computing and use Gmail and other Google apps for most of their productivity needs. What’s been lacking is a really good Chrome-based laptop, especially for software developers and other power users who demand speed and high-resolution displays. “In my personal experience, it’s the fastest laptop I used,” Pichai said.

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As an additional enticement — or perhaps to make up for its limitations — the Pixel comes with a whopping 1 terabyte of free storage on Google Drive, enough to store most people’s photo and music collections many times over, for 3 years. In a couple of months, Google will also offer a Web-version of Quickoffice, which Google acquired last year, and which will provide better compatibility with Microsoft Office programs than Google Docs does currently.

The Pixel also represents Google’s latest foray into hardware. Unlike prior Chromebooks, which were made by the likes of Samsung, Acer, and Hewlett-Packard (HPQ), the Pixel is entirely a Google product, though it is being assembled by a contract manufacturer in Asia.

All indications suggest the transformation of Google from an Internet company into a computing company that makes both hardware and software will continue apace. On the hardware front, there will be Google Glass, of course, and more Nexus-branded mobile devices made by Google or its Android partners. There will also be handsets and tablets, made by Motorola Mobility, which Google purchased for $12.5 billion. In the words of Larry Page, it’s all about keeping up with the unprecedented rate of change in personal computing. “It’s why we’ve put so much focus on devices,” Page said during the company’s most recent earnings call. “They’ve been one of our biggest bets in the last few years — along with the software to go with the devices, Chrome and Android.”

It will be interesting to watch. If it succeeds, the Pixel would be a huge validation for Google’s cloud-only vision of computing. The company has long hedged its bets with Android, which runs apps whether they live in the cloud or on the device itself. But no one in the industry would have more to gain if pure cloud computing becomes a truly viable alternative to the PC.