The average of seven consecutive positive integers is 126. The smallest of these integers is:

Let the 7 consecutive positive integers be $x,x+1,x+2,x+3,x+4,x+5,$ and $x+6$.
Their average is the middle term, which is the 4th term:

$$\text{Average}=x+3$$

Given, $x+3=126$

$$x=126-3=123$$

Therefore, the smallest integer is 123.