The Rio Grande technically met its end on 13-Oct-1988, when Rio Grande Industries took control of Southern Pacific Transportation Company and assumed the larger road's name. From there on, it was technically Southern Pacific. However, most of us don't really think of this so much as the end, since business largely continued as usual on the former DRGW system for several years afterwards. Also, as a technical point, the Rio Grande had purchased the Southern Pacific and remained - it just assumed the latter's name for railroad operations. The real end started on 11-Sep-1996, when Union Pacific was granted control over the Southern Pacific empire (SP, SSW, and the D&RGW). A little less than a year later, on 30-Jun-1997, the Denver & Rio Grande Western Railroad formally ceased to exist, even on paper.
Even though the Rio Grande as a corporation has passed into history, the railroad itself lives on. The majority of the ex-DRGW standard gauge system is now operated by Union Pacific?, including Denver-Salt Lake City via the Moffat, Tennessee Pass (currently mothballed), the North Fork, Craig, Sunnyside, Montrose, and Cane Creek Branches, and the Joint Line.
The two major pieces that have been sold to other parties are the Aspen Branch and the Walsenburg-Alamosa line, including the San Luis Valley branches (Antonito and Creede).
The Walsenburg-Alamosa line, as well as the Antonito Branch and the Creede Branch (from Alamosa out to Derrick, CO, just beyond South Fork) were sold on 29-Jun-2003. The San Luis & Rio Grande Railroad, a RailAmerica subsidiary, took over from Union Pacific. They're based in Alamosa, and run the primary train over Veta Pass during the night.
The remainder of the Creede Branch from Derrick, CO, to the end of track at Creede, CO, was purchased by the Denver & Rio Grande Historical Foundation and are intended to be restored as a tourist hauler.
The Aspen Branch, from Glenwood Springs, CO, up the Roaring Fork Valley towards Aspen, was purchased by the Roaring Fork Transit Authority with the intended purpose of preserving the corridor for rail service and possible future light rail use. Recently, they've decided to abandon this idea and pull the rails to make way for a trail.
What's left of the narrow gauge system was dismantled or sold off long before the end of the Rio Grande, with the mergers of the last three decades having no effect on the 3-foot parts of the system. The last real Rio Grande narrow gauge operations ended in the late 1960s, when the majority of the San Juan Extension was abandoned. From the main line between Alamosa and Durango, 64 miles was saved in 1970 by the states of Colorado and New Mexico as the Cumbres & Toltec Scenic Railroad. In addition, the Rio Grande kept the now-isolated Silverton Branch from Durango to Silverton, as it had acquired a healthy tourist business. Even that was eventually sold, and the narrow gauge Rio Grande ceased to exist entirely in March of 1981, when the Durango & Silverton was created from the Rio Grande's Silverton Branch tourist operations.