We present the discovery of GRB 020405 made with the Interplanetary Network (IPN). With a duration of 60 s, the burst appears to be a typical long-duration event. We observed the 75 arcmin$^2$ IPN error region with the Mount Stromlo Observatory's 50 inch robotic telescope and discovered a transient source that subsequently decayed and was also associated with a variable radio source. We identify this source as the afterglow of GRB 020405. Subsequent observations by other groups found varying polarized flux and established a redshift of 0.690 to the host galaxy. Motivated by the low redshift, we triggered observations with WFPC2 on board the Hubble Space Telescope (HST). Modeling the early ground-based data with a jet model, we find a clear red excess over the decaying optical light curves that is present between day 10 and day 141 (the last HST epoch). This bump has the spectral and temporal features expected of an underlying supernova (SN). In particular, the red color of the putative SN is similar to that of the SN associated with GRB 011121 at late time. Restricting the sample of GRBs to those with z<0.7, a total of five bursts, red bumps at late times are found in GRB 970228, GRB 011121, and GRB 020405. It is possible that the simplest idea, namely, that all long-duration γ-ray bursts have underlying SNe with a modest dispersion in their properties (especially peak luminosity), is sufficient to explain the nondetections.