This is an interesting question, and with all that's been written about the sign, since it was first installed in 1959, we thought it would be straightforward locating the answer. Wrong!
While numerous sites we came across referenced how the sign has apparently "moved several times" since it was first installed, none of them offered any information as to where it was originally placed, nor when or to where it was moved. For example, here's an excerpt from a recent article in the L.A. Times. "In time, the sign became somewhat of a problem child for Las Vegas civic fathers, enduring more than one demolition attempt to make way for a more modern roadside greeting. The sign has been moved several times, deployed farther and farther south to remain on the outskirts of the latest casino development." Okay, but where are the details?!
Finally, we placed a call to the Neon Museum downtown, and spoke with a helpful member of staff who had previously been a Las Vegas tour guide. His recollection was that the sign was originally located close to Tropicana Avenue, but that when the Hacienda was built further south, the sign was moved so that it was still encountered before the first hotel (now Mandalay Bay, built on the site of the Hacienda).
However, they broke ground on the Hacienda in '55, with the casino finally opening in '57, i.e., two years before the Las Vegas sign was installed, so this timeline doesn't make sense. The Tropicana opened in '57, but construction was already underway on the Hacienda, located a mile south of the nearest casino at that time, which was the Flamingo, so why would the sign have been originally located anywhere but to south of the Hacienda site? It makes us wonder whether the sign did actually ever move, or if it's just an urban myth dreamt up by people assuming it must have moved, seeing as the city has expanded so fast. From the earliest image we have of the famous sign, it's impossible to judge where it's located, since there's really nothing, especially any surviving landmark, anywhere near it.
If anyone out there can shed any additional light on this mystery, we'd love to hear from you. Below is the earliest photograph we've seen of the sign, taken in the 1960s. With no landmarks in the vicinity, it's really hard to gauge the location.