The Discone antenna is remarkable in that it is capable of receiving and transmitting over a wide range of frequencies with good matching. Because of this, it is a good match for SDR receivers such as the popular RTL-SDR sticks.
The only really tricking thing about making a discone is that the disc has to be balanced at the very top of the cone, which is mechanically awkward.
The two parts here allow the cone to be solidly clamped and provide an adequate base for the disk. There also two holes for bring the coax centre and braid out to the disc and cone.
The base part has a socket at the bottom for 25mm (1 inch) plastic conduit for mounting
This antenna illustrated is designed for 400MHz and up, and as such transmits well on the 70cms amateur band, US and UK PMR channels and 23cms. It also receives aircraft ADS-B signals very well. I used .3mm plastic sheet for the cone and 2mm plastic for the disc, and then covered them with aluminium weatherproof tape. Be sure to check for continuity across the tape stripes.
The screenshot is of a calculator by VE3SQB which can be downloaded from http://www.ve3sqb.com/ if you want to make attenna's for other ranges.