Just to add to the already-great answer, please see attached photo of the relevant Mk2 handbook page (I couldn't be bothered getting my scanner working - Windows Vista - hope you understand

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I'm not sure why you are bothering to change the stereo for someone else, mind

But I do know what you mean - I hate the silvery junk that people have with the tiny buttons and the tendency to drain the battery unless you pop the face off.
Going way off topic now, in my 164 I have a Chinese stereo that I bought directly from China via someone in NZ. Price was US$99 and I paid way over that (I got the invoice enclosed in the shipment - LOL). Anyway on paper it's amazing - motorised 4" LCD in the single-DIN unit, TV, DVD player, reversing camera input, Bluetooth interface to mobile phone, plays MP3s of course, USB port, RDS radio...
... but what this doesn't tell you is that the touchscreen (for doing things such as next-track, radio station etc.) uses TINY squares of green overlaid on the supermarket-DVD-player graphics, so it's IMPOSSIBLE to use while driving (they give you a little stylus, as if...) - it lets out horrid beeps like a PC speaker, the volume control is not-linear, the sound quality is bad, and it lights up with BLUE LEDs :yuck: - AND, this is the worst one, at night there's no dimming, so the whole LCD lights up in blue (if you're playing MP3s or listening to the radio) - it lights up the interior of the car. The only way to solve that is to turn it off or play a DVD (soooooo much less distracting

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...so I'm on the verge of picking up an old-style Blaupunkt with CD Changer and Auxiliary Input... because that will not only look much better, it will sound better, it will be easier to use (or, possible to use on the move without selecting 3mm-squares with a stylus) and the only downside is that I won't be able to watch DVDs or use my reversing camera (which I haven't hooked up yet anyway) - TV reception is hopeless in a car, by the way, and the Bluetooth sound quality was poor.
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A little more trivia.
The more deluxe models of the Mk1 Uno - i.e. the ones that came with a cigarette lighter - have a brown power wire for the stereo! And, an antenna on the roof or the door pillar - with a cable! That's the limit of the factory wiring/stereo...
As far as the choice of radio is concerned... in NZ, a number of old Unos have AWA/Clarion radio-cassette head units: 45/60 have the type with two knobs, 70SL has the type with two knobs and digital tuning. Most of these actually have a 'FIAT' label on the cassette flap. This may have been something created by the importers.
Later models (Mk2) and Tipo models tend to come with a digital-tuning Blaupunkt radio-cassette which is probably the same as what Chas (1986Uno45S) mentioned. This is not the two-knob (shaft-mounting) type, so instantly looks more modern, is less prone to rattles, and of course the digital tuning is more effective.
I think this was the start of a long tie-up between FIAT and Blaupunkt. Basically, Blaupunkt is a Bosch product (they carry a Bosch part number). There are cheap ones and expensive ones. Anything from, say, the Mk2 Punto onwards tends to have an integrated 'cheap' Blaupunkt unit fitted. The expensive Blaupunkt units have unfortunately disappeared off the market, as Blaupunkt now seem to compete at entry-to-mid level with Sony, Pioneer etc. and also as they provide the standard-fit stereo for mid-range cars.
The last expensive Blaupunkt unit I bought was a 'London RDM168', and had I known that it is now impossible to replace, I would have kept it rather than giving it away in the car when I sold it
The London RDM168 (yes, I would love to buy another, if anyone has one) was something of pure class compared with the rubbish on the market today... The piano-black fascia carried a small matrix LCD at the centre, in sensible instrument-like shades of green, yellow, and red. It had night dimming to a preset level (1-9). There was a round knob for volume, with a power button (yes, sadly I think the power button was blue, but I'll let them have that, given the name). The other side carried a round up-down/left-right control of great tactility. Remaining buttons were the trademark rounded black shape with white plastic moulded-in. There were eight 'soft'-buttons - buttons whose function matched what was on the LCD - e.g. the names of radio stations, or the settings for the CD that was playing - a bit like an ATM machine has. It was thus brilliantly simple to use, with only the minimum of buttons yet so many nice features...
...such as a parametric equaliser, sound that quietly increased or decreased (no loud blast of volume), the power that can be turned on with the key off, yet will still turn off when the key goes off, I mentioned the night dimming, the soft 'bong' sound after you turn it off to remind you to remove the keycard (I never did), the pleasant tock-sounds with the buttons...
...it had an auxiliary input, which meant that I connected my iPod to it - in 2003, this was a long time before many people thought of doing this.
...because the face folded down to insert the CD, that made the face as large as possible - it wasn't detachable, so it was solid too.
...it had a feature that I could never use, because we don't have RDS (properly) in NZ (one station uses it to broadcast song names and that's it) - if you have traffic information, this London RDM168 could actually stay tuned to the traffic information while the car was parked (and the radio switched off) - and it would MEMORISE the traffic information to replay it when you turned the radio on later and pressed the 'TM' button... and THAT seems like an amazingly-cool feature.
So, you didn't get Bluetooth, TV, DVD, video-in, or GPS for your 400 pounds back then in 2002 (I paid a little less), but you did get something of real quality that makes even factory-integrated systems feel cheap.
I just realised that after all these years, I mis-remembered the model (I think), as a Google search suggests it's more likely to be London RDM169. One sold for peanuts in Austria:
http://cgi.ebay.at/BLAUPUNKT-London...0589395QQihZ008QQcategoryZ14259QQcmdZViewItem Let's hope I can find another one day.
Cheers!
-Alex