#Smalltalk (now seriously), 123#
Sorry for answering twice, but consider this a serious answer, while the other one was more like humor. The following is actually executed right at this very moment in all of our machines (in hardware, though). Strange that it came to no one else's mind...
By combining two half-adders, and doing all bits of the words in parallel, we get (inputs a,b; output in s) readable version:
s := a bitXor: b.
c := (a & b)<<1.
[c ~= 0] whileTrue:[
cn := s & c.
s := s bitXor: c.
c := cn<<1.
c := c & 16rFFFFFFFF.
s := s & 16rFFFFFFFF.
].
s
The loop is for carry propagation. The masks ensure that signed integers are handled (without them, only unsigned numbers are possibe). They also define the word length, the above being for 32bit operation. If you prefer 68bit addition, change to 16rFFFFFFFFFFFFFFFFF.
golf version (123 chars):
[:a :b||s c n|s:=a bitXor:b.c:=(a&b)<<1.[c~=0]whileTrue:[n:=s&c.s:=s bitXor:c.c:=n<<1.c:=c&16rFFFFFFFF.s:=s&16rFFFFFFFF].s]