27kmiles. Runs great. Engine cuts out.

Mate I use to drive a freightliner and it cut out on me one day, I eventually got it started but cut out again after about 5 minutes, I rang a mechanic to come out and he found a heat sensor wire came off the radiator and that caused the motor to cut out... Hope I haven't wasted your time.......

Does the check engine light come on?

This happened to my 95 Royale while driving around town. I checked the battery terminals and one was slightly loose. After tightening it, no more dying.

i had the exact same thing happen to me with my first car, a 1966 mercury comet.

i replaced the fuel pump, rebuilt the carburetor (twice), and finally rebuilt the entire engine! still, every time afterwards, the car ran ok, then would cut off again!

it turned to be just a $2 clogged fuel filter. i' suspect the same with your vehicle. after removing the filter, simply blow through it. any resistance, means it needs replacing.

that is very good, the cars momentum will in fact restart the car.
with a M/T tranny, with A/T it slips. so is car a M/T , stick?
do post tranny type, we cant guess that....???
misfires and stalls. the engine stalls, the ECU tries very hard to
prevent stalls, but failed to do so...
you are loosing electric power, 12vdc , and this kills EFI
is the CEL lamp not coming on at 0rpm when the engine does stall>?
if not you lost 12v. hard fact.
this is not hard to find or fix ,if lost 12v.
when you lose 12v the dash goes dead, warning lamps fail.
and in such a stated, the key crank will be dead.
after all there is no 12v to do the cranking, from key.
its an old car and all cars fail there is no car that does not.
all machines do. even space shuttles. fail 1:50 times.

if you have a voltmeter, connect it to the CIG lighter port.
and see if the 13.3 to 15vdc running drops to zero or below 10v.
if yes, BINGO.
the CIG steals power from the ignition power feed.

when any key start fails to crank here are some tricks.

in view of your post , you have 2 non related problems
or the battery cables are corroded. ever clean them>
one other trick is have head lights on, at the stall
are the headlamps FULL bightness at the stall>?????? very easy test.

http://www.fixkick.com/nocrank.html#tricks

SINCE CAR WONT CRANK. NOW IS THE TIME TO REPLACE THE IGNITION SWITCH KEY LOCK CYLINDER ITS BROKEN THAT WHY THE CAR KEYS WOULDNT TURN OFF CAR.DISCONNECTING BATTERY CABLES WITH CAR RUNNIMG WILL.SHORT OUT ALTERNATOR AND PCM.IF IN CASE YOU HAD TO REMOVE BATTERY CABLE REMOVE THE GROUND NEGATIVE CABLE FIRST.MAKE SURE ALL ELECTRICAL COMPONENTS OFF LIKE RADIO. AC SYSTEM OFF.THIS WILL PREVENT LARGE ELECTRICAL ARC FROM BATTERY THIS IS ONE TIME PROCEDURE IN CASE ENGINE WONT SHUT OFF ITS A GAMBLE BECAUSE IT WILL TAKE OUT ALTERNATOR AND PCM.REPLACE THE SWITCH AND KEY LOCK CYLINDER DONT KEEP STARTING CAR BY DISCONNECTING THE BATTERY CABLE YOU WILL DAMAGE IGNITION ELECTRICAL SYSTEM.

check your brake switch. if high resistence it can cause override for autostart.

Your oil is a little low, and the timing may be off a bit but if it smooths out after warm it is probably ljust
low oil.

Check your RPM sensor.  Mine did virtually the same thing.  It was NOT the fuel pump like Audi dealer said-after I repaired it.  

Take both cables of the battery and clean them with a wire brush, reinstall them making sure those little 5/16 hex bolts are nice and tight in the battery. Trace the battery ground cable back to where it bolts to the engine block, make sure it is very clean and tight. There should be a body ground somewhere, this is a conductor between the engine block and the metal body part of the engine compartment- clean and tight as well. If you can't find the body ground, make one.
Now get on your hands and knees and inspect behind both front kick panels and under the dashboard valence, inside the car. You are looking for single or clusters of brown or black wires that are bolted to the body inside the car. Clean and tight for these as well. I promise, this is where your problem lies, but I'm betting on loose battery terminals right now.