Or you may use the base
function ave
df[df$Distance == ave(df$Distance, df$Record1_ID, FUN = min), ]
# Record1_ID Record2_ID Distance
# 2 6 11_Bla 0.8525580
# 8 684 81_Lun 0.9148743
# 10 3065 137_Pro 0.8957428
# 11 3065 138_Rec 0.8957428