AN OLD ATIS; CLRNC TO A CLOSED RWY; A LAST MIN APCH CHANGE; A LATE NIGHT ARR; AND DETERIORATING WX AT KLAX RESULT IN A HEAVY CREW WORKLOAD.

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IMPROPER ATC HANDLING. LATE NIGHT ARR INTO LAX; VERY LITTLE TFC AND LATE HANDOFF FROM LAX CENTER TO SOCAL APCH. EVENTS BEGAN 10 OUTSIDE CIVET AT FL190. WE WERE TOLD TO DSND VIA THE CIVET 4. ATIS WAS OVER 1 AND 1/2 HRS OLD. OLD ATIS STATED CLR CONDITIONS AND 9 MILES VIS. LOOKING OUT THE WINDOW; IT WAS OBVIOUSLY LOW OVERCAST. SHORTLY CLEARED FOR THE ILS RWY 25 L APCH. ABOUT 3 MILES OUTSIDE FAF; (LIMMA) WHILE RUNNING THE FINAL DSCNT CHKLIST; APCH CLRED US FOR THE VISUAL RWY 25 RIGHT. I STATED; 'UNABLE... YOUR ATIS IS INCORRECT; WE HAVE AN OVERCAST LAYER AND NEED THE ILS.' AFTER A SHORT PAUSE; SOCAL CLRED US FOR THE ILS RWY 25 LEFT; SIDE-STEP RWY 25 RIGHT. ABOUT 3 SECONDS AFTER THAT CLRNC; WE LOST ALL ILS GUIDANCE (WANDERING NEEDLES; LINES THROUGH G/S AND LOC; AUTOPLT WARNINGS; ETC). WE QUICKLY REALIZED THAT THE ILS RWY 25 LEFT HAD BEEN TURNED OFF! CAPT UNCOUPLED AND HAND FLEW WHILE I TUNED AND IDENTIFIED RWY 25 RIGHT. SINCE WE WERE CONFIGURED AND STABILIZED BY THE FAF; WE CONTINUED THE APCH. AFTER LNDG; I STERNLY QUERIED ATC ABOUT THIS EVENT. I AM WELL AWARE THAT SOCAL AND LAX SWITCH RWYS AND STARS OFTEN; BUT SWITCHING RWYS; TURNING OFF THE ILS THAT IS IN USE WITH AN ACFT AT 1500 FT NEAR THE FAF IN IMC IS EXTREMELY DANGEROUS! CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR NOTED THAT RWY 25L HAD A LARGE LIGHTED 'X' WITH MEN AND EQUIPMENT ON THE RWY INDICATING CLOSURE WHEN THEY LANDED. THIS WAS NOT RPTED ON THE ATIS OR ANY PREVIOUSLY RECEIVED NOTAM AND HE'S CONCERNED WHY NOTHING WAS SAID BY EITHER THE TWR OR APCH. RPTR ALSO INDICATED THAT THEY BROKE OUT OF THE OVERCAST AT APPROX 700 TO 800 FT.