where the heroine made the mistake of committing adultery (the scarlet sin of that story) and was sentenced to wear a big red A on her clothing for all time, to identify her as an adulteress, which you can imagine ruined her life... one mistake, a life sentence among your peers.

The "scarlet sin" is adultery

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Note added at 6 mins (2005-03-23 17:56:02 GMT)
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http://www.wayoflife.org/fbns/scarletsin.htm

and

Isaiah, in his day, said: \"Though your sins be as scarlet,
they shall be white as snow, though they be red like crimson, they shall be as wool\" (Isaiah 1:18).
Adultery is sometimes referred to as the scarlet sin.

at, for example,
http://www.acts2.com/thebibletruth/The_Nicolaitanes.htm


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Note added at 6 mins (2005-03-23 17:56:21 GMT)
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http://www.wayoflife.org/fbns/scarletsin.htm

and

Isaiah, in his day, said: \"Though your sins be as scarlet,
they shall be white as snow, though they be red like crimson, they shall be as wool\" (Isaiah 1:18).
Adultery is sometimes referred to as the scarlet sin.

at, for example,
http://www.acts2.com/thebibletruth/The_Nicolaitanes.htm

Same as the sin in the "Scarlet letter"

Married party has sex with person NOT their spouse.

If the given text had written "the scarlet sin", I would have agreed with the above interpretation of "adultery".

But since the text says "some great scarlet sin", I think it must mean a deadly sin, an ultimate sin or something close. (We must also consider the nature of the original text - named as "classics of Gospel Truth", which I detail at the bottom). Some linked examples:

Isaiah 1: 18 Come now, and let us reason together, saith the Lord: though your sins be as scarlet they shall be as white as snow; though they be red like crimson, they shall be as wool.

"Persecution of the saints of God is a scarlet sin, that calls aloud to heaven for vengeance."
http://www.spurgeon.org/sermons/3505.htm
A Miracle of Grace - A Sermon Delivered by C. H. SPURGEON, Published on Thursday, March 30th, 1916.

"Again, thirdly, the knowledge of the law binds the person so much the more to obedience, by how much the more he knows it; ... and so, when you sin not knowing the law, the sin is the same for substance it would be if you had known it, yet knowledge dyes it, makes it a 'scarlet sin,' as Isaiah speaks, far greater deeper in demerit than the sin itself...
http://www.newble.co.uk/goodwin/aggsinkno.html
AGGRAVATIONS OF SINNING AGAINST KNOWLEDGE.

"We know that the one scarlet sin, beside which all the cambric sins we have often denounced are too trivial to be interesting, is the sin of human exploitation."
http://www.bestsermons.net/1926/The_Gospel_in_the_Present_Ag...
THE GOSPEL IN THE PRESENT AGE - CARL S. PATTON, D.D.

(Note, I do not read "human exploitation" above as just prostitution.)

"There is no scarlet sin which has been omitted, for the Lord has said, "Come now, and let us reason together: though your sins be as scarlet." He entered into a covenant that he would not turn away from us, to do us good; and no circumstance has arisen, or can arise, which was unknown to him when he thus pledged his word of grace."
http://www.blueletterbible.org/Comm/charles_spurgeon/sermons...
Perseverance in Holiness, A Sermon Delivered on October 6th, 1889 by C. H. SPURGEON

***
Finally, I think the second and third sentences (and also the first sentence, but slightly changed) of the question are quoted from John Charles Ryle, the First Bishop of Liverpool (1816-1900), because I found the below text in a document:
"Reader, I know not what your state of soul may be. But I warn you to take heed that you do not perish for ever for want of striving. Do not suppose that it needs some great scarlet sin to bring you to the pit of destruction. You have only to sit still and do nothing, and you will find yourself there at last..."
http://www.tracts.ukgo.com/strive.doc
J. C. Ryle Tracts

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Note added at 1 hr 55 mins (2005-03-23 19:44:52 GMT)
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Notes regarding:
Isaiah 1: 18 Come now, and let us reason together, saith the Lord: though your sins be as scarlet they shall be as white as snow; though they be red like crimson, they shall be as wool.

\"What does this mean to you?

It means this: No matter what sin you may have committed: prostitution, idolatry, adultery, murder, theft, blasphemy or sedition: you can be forgiven - if you want forgiveness...\"
http://avoiceinthewilderness.org/salvation/sermon74.html

And:

\"The word \"scarlet\" means to shine. It denotes a bright red color due to its appearance and splendor. It derives its hue from a worm or insect which can be found on the leaves of a shrub of the ilex group. It also carries the meaning of something being double-dyed, done twice, or twice-dipped.

In speaking to the people of Israel, God chose the word \"scarlet\" to depict the stain of sin as being a deep-seated, ingrained, fixed, definite, lasting, and permanent color that nothing - dew, rain, washing, or long usage - could ever remove. No effort of man could take this stain away.\"
www.walkingintruth.org/Walkingintruth/devotions/wit2003/WIT...