I write an awful lot about one-hit wonders on this site, but near as I can figure, no act has ever been a pure two-hit wonder: two songs going to Number One without anything else even reaching the Top Forty. The closest was the German disco act Silver Convention, which sent "Fly, Robin, Fly" to the top spot on November 29, 1975, then had "Get Up and Boogie (That's Right)" stall out at Number Two for three weeks the following spring. The followup, "No No Joe," peaked at Number 60, and Silver Convention was done making hits.
"Fly, Robin, Fly" won a Grammy for Best R&B Instrumental, even though, as you'll probably recall, it had lyrics. The song had a total of six words, as a matter of fact; they were, in alphabetical order, "fly," "robin," "sky," "the," "to" and "up." "Get Up and Boogie (That's Right)" also had just six words, all of which are helpfully listed in the title, so I don't need to repeat them here. I don't know how many words "No No Joe" had.
There are a couple of footnotes to this feat. The Cuban bandleader Perez Prado had two Number One singles ("Cherry Pink and Apple Blossom White" and "Patricia") and no other hits during the rock era, but both those hits predate the Hot 100. "Cherry Pink" dates back to March of 1955, which is as early as my reference materials go, and I rather suspect that Prado had other Top Forty hits before 1955. He also wrote "Mambo No. 5," which Lou Bega turned into a hit in 1999.
Wayne Fontana and the Mindbenders sort of matched Silver Convention's record. Their "Game of Love" went to Number One on April 24, 1965, and the next spring, the Mindbenders (note new band name) had another huge hit with "Groovy Kind of Love," which went to Number Two. But in between those two hits, Wayne Fontana left the group - in the middle of a show, from what I've read. He was replaced as lead singer by guitarist Eric Stewart, who would later form 10cc. Anyway, whether the two Mindbenders' hits are by the same band is a decision the reader will have to make.
A Taste of Honey went to Number One with "Boogie Oogie Oogie" in 1978, and then to Number Three with "Sukiyaki" in 1981. "Sukiyaki" did go to Number One on both the R&B and adult contemporary charts. Plus, it's better than anything Silver Convention ever did.