Are we born sinners? We can be quite certain that God didn't originally create us as sinners. Rather Adam and Eve chose sin in the Fall.

The doctrine of original sin is widely held in Christian churches, but those opposing it are not a small minority, either. As original sin means that we inherit our sinfulness from the Fall, I think it might be equivalent to the question asked here: Are we born sinners?

Please provide at least the biblical evidence for an answer -- non-biblical arguments are, of course, welcome as additional information. Also, I might be wrong, and some believers in original sin might not think we're born sinners. If these notions are actually separate, please elaborate in the answers.
Since I had to find relevant Bible verses for another purpose, I'll answer my own question with them. I still expect someone to write a more full answer. This interpretation/selection of passages is close to the Lutheran view.

David does seem to state that we're born sinners:


  Psalm 51:5 (ESV)
  Behold, I was brought forth in iniquity,
  and in sin did my mother conceive me.


Paul also writes to the Romans, saying that "one trespass led to condemnation for all men"


  Romans 5:12-13,18 (ESV) 
  12 Therefore, just as sin came into the world through one man, and death through sin, and so death spread to all men because all sinned— 13 for sin indeed was in the world before the law was given, but sin is not counted where there is no law.
  
  18 Therefore, as one trespass led to condemnation for all men, so one act of righteousness leads to justification and life for all men.


He also makes a similar statement when writing to the Corinthians:


  1 Corinthians 15:22 (ESV)
  For as in Adam all die, so also in Christ shall all be made alive.


The Bible does seem to say that we indeed are sinful from birth because of the Fall of Man.