David French said:
The Gene Kelly episode was the last to be recorded but AFAIA it wasn't the last to be aired. It all depends on what you determine the last episode to be - the "official" episode list has the Roger Moore episode as #120.
The last one broadcast in the US appears to have been Melissa Manchester, on 8 June 1981. The last one aired in the UK was Shirley Bassey, on 15 March. Gene Kelly aired on January 4 in the UK, and February 28 in the US. Roger Moore, on the other hand, opened the fifth season in the UK.
(To qualify the above: I don't know which region/station the above US dates are for; I suspect either NY or LA. The UK dates are AFAIK the network dates, although it's possible that one or more regions either opted out or ran the shows on different dates. UK tx dates for TMS are a bit odd, especially in the early days before the show took off...!)