Changing
your speedometer cable
Vespa speedo cables snap. All the time. I have no idea why - it
doesn't seem to be an intrinsic design flaw. Just the limitations
of a mechanical system I guess. Let me start by saying if you actually
want an accurate speedo on your Vespa, chuck out the existing
one and buy a Sigma Sport BC800 Bicycle Computer. If you're happy
with
the speedo in there and just need it to work at all, then read
on.
The first thing you need to do is remove the speedo. To do this,
take the headlight out (see this article
if you don't know how),
and if it is still present, remove the screw on the underside of
the headset that holds the speedo in. If the speedo cable stays
attached you should be able to lift the speedo enough so that the
cable can be unscrewed or unclipped from the underside of the speedometer.
If you can't lift it enough, try pushing the cable up into the
fork from the botttom. That should give you enough leeway. Here's
the headset with the light and speedo hanging out. 

As you can see the speedo bulb connection is still
attached and the speedo is hanging by it. Now you need to extract
the inner cable. If it has snapped at the bottom then you can
pull it out from the top. You do need to get the bottom
end out though. Unscrew the fat bolt in the suspension arm:

You should be able to extract the cable end and
speedo gear with some needle-nosed pliers:

If you have trouble you can remove the large rubber
bung in the suspension arm for another angle of attack. The speedo
drive gear (which in this case still has a length of rusted cable
through it) looks like this.

It looks even more impressive when it's clean.
(The other thing is just the removed screw from the suspension
arm.)
To replace the outer cable you need to undo this
bolt here:

It is a complete pain to get at, but with the right
spanner and small hands it is possible. Undoing it frees a small
plate which you can slide up the speedo cable and this allows
you to pull the cable out of the hub:

To get the new outer on you need to feed your new
cable inner up the old outer from the bottom: 
Hold this in place while you slide the old outer
up from the top over the inner, leaving the inner in place
and allowing you to slide the new outer down the new inner
and out to the hub. Pull the inner out and feed it through
the right way round from the top to check the lengths. You
may need to trim the outer to size: ideally you need the inner
to extend from the outer slightly less than the length from
the point where the outer fits into the suspension arm to the
fat bolt in the end of the arm.
Once the outer has been cut to size remove the
inner, grease it thoroughly and reinsert it. Put the plate
on the new cable and feed it into the suspension arm. Tighten
the bolt on the plate and slide the speedo drive gear up the
inner cable from the end of the suspension arm, turning the
front wheel by hand to encourage the gears to slot into place
correctly. When you spin the front wheel the cable inner should
itself spin (visible at the headset). If it's all correct replace
the large bolt in the end of the arm (and the rubber bung if
you
removed
it).
With everything secure at the wheel feed the
cable up the fork as much as possible to give yourself enough
room to clip or screw the speedo onto the cable. If the cable
is seated in the speedo properly and the drive gear is seated
properly the speedo needle should rise when you spin the
front wheel.
Replace the speedo in its hole and pull the cable
back down at the wheel end to pull the speedo snugly into place.
If you want to you can put that screw back into the underside
of the headset. I threw mine away. Now just replace the headlight
and the job is done.

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