* U.S. employers shed jobs but unemployment rate steady * Euro poised for biggest wkly loss vs dlr in its lifetime * U.S. House vote on rescue plan awaited (Recasts, updates prices, changes byline, changes dateline, previous LONDON) By Nick Olivari NEW YORK, Oct 3 (Reuters) - The dollar continued its rally against the euro on Friday after a report showed the unemployment rate remained steady in September, even while U.S. employers shed jobs. U.S. employers cut payrolls at the steepest rate in 5-1/2 years last month, according to a Labor Department report, slashing an unexpectedly large 159,000 nonfarm jobs as employment contracted for a ninth straight month. But foreign exchange investors focused on the unemployment rate which was unchanged from August at 6.1 percent. The report added to the optimism surrounding the dollar which was already on track for its best weekly gain versus the euro in the single currency's lifetime after the European Central Bank opened the door to rate cuts on Thursday. "U.S. non-farm payrolls are slightly worse than expected, with a net revision to prior months... along with the unemployment rate holding steady, which somewhat mitigated the weakness," said Brian Dolan, chief currency strategist, at Forex.com in Bedminster, New Jersey. "The market is not showing any sign yet of abandoning the strong U.S. dollar theme of the week. I remain optimistic that we will get a TARP passage, ugly as it may be and the market will embrace risk as an antidote to fear." In early New York trade, the euro fell 0.4 percent on the day to $1.3758 EUR=>, having plumbed a 13-month trough of around $1.3743 on Thursday. The single currency was on track for its worst weekly percentage loss since its inception in 1999. Against the yen, the single currency was down at 144.71 yen EURJPY=>, but held above a two-year trough of around 144.56 yen, according to electronic trading platform EBS EURJPY=EBS, touched earlier in the global session. The dollar eased 0.2 percent to 105.16 yen JPY=>. The dollar index .DXY>, which gauges its performance against a basket of six major currencies, was up 0.3 percent at 80.704 .DXY>, within easy reach of the 13-month peak of 80.794 hit in the previous session. The dollar index, up 4.448 percent this week, was on track for its best weekly gain since September 1992 at current prices, according to Reuters data. The dollar has surged this week as the international nature of the financial market crisis was highlighted by the rescue of some major European lenders including Fortis this week. "The U.S. authorities are extremely committed to solving this crisis...Europe would never be able to some up with such a huge stimulus package as they are talking about in the U.S. and that puts the euro area at a very high risk," Danske Bank senior currency strategist John Hydeskov said in London. The deteriorating financial backdrop prompted the ECB, which left rates unchanged at 4.25 percent on Thursday, to open the door for its first rate cut in more than five years with President Jean-Claude Trichet saying inflation risks have eased as financial market turbulence hit the euro zone. "Trichet opened up for a rate cut, some would say that it's a bit late, but they (ECB) needed some time to change the mood of the board," Hydeskov said. The squeeze in interbank lending -- which has driven three-month dollar Libor rates USD3MFSR= up a full percentage point in two weeks to more than double the Federal Reserve's 2 percent rate target -- is a major factor behind the dollar's gains, analysts say. Against that backdrop, a slew of data this week showing the U.S. economy has likely fallen into a full-blown recession has done little to take the wind out of the dollar's rise, even as the Fed is seen likely to cut rates as well this month. (Additional reporting by Veronica Brown in London,Editing by Walker Simon)) (Reporting by Nick Olivari and Gertrude Chavez-Dreyfuss)

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