In the bowl of a stand mixer fitted with the dough hook attachment, whisk together the warm water, honey and yeast until combined. Let it stand for 5 minutes, until very foamy.
To the bowl with the yeast mixture, add both of the flours, oil, caraway seeds and salt. Using a wooden spoon or rubber spatula, mix the ingredients together until all of the dry flour from the bottom and sides of the bowl is combined and a shaggy dough forms.
With the stand mixer, mix on medium-low speed for 6 to 8 minutes until most of the dough balls up around the dough hook and it is smooth and slightly sticky.
Place the dough in a clean, lightly oiled bowl. Flip the dough over once so that the top also has a coating of oil. Scrape down the sides of the bowl to shape the dough into a ball. Cover the bowl with plastic wrap and let it rise in a warm, draft-free spot (75°F) for 30 to 45 min, until doubled in size.
Meanwhile, grease a 9x5-inch loaf pan with oil or baking spray. Set it aside.
Turn the dough out onto a very lightly floured surface. With moist hands, stretch and shape it into a 9-inch square. Using a bench scraper, coax the dough from the surface to roll it into a log. Pinch the seams on the sides together tightly to seal and gently tuck them under the loaf.
Place the dough in the prepared loaf pan seam side down. Brush a piece of plastic wrap lightly with oil or spray with nonstick spray. Cover the loaf pan and let it rise in a warm, draft-free spot (75°F) for 35 to 45 minutes, or until the dough rises ½ to 1 inch above the top rim of the pan.
After 15 minutes of the second rise, preheat the oven to 350°F.
Bake the bread for 35 to 40 minutes, until the top of the loaf is golden brown and it sounds hollow when tapped. Let it cool in the pan for a couple of minutes, then turn it onto a wire rack to cool completely before slicing.
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Notes
Use molasses: In place of honey, use molasses for deeper color and brown sugar flavor notes.
All-purpose flour versus bread flour: You can use all-purpose flour in place of bread flour, if that is what you have on hand. However, because rye flour tends to have less gluten than wheat flour, it's helpful to use a higher-gluten wheat flour, like bread flour, instead of a lower-gluten wheat flour like all-purpose flour.
Do not add too much flour, even if the dough seems sticky. This dough tends to be stickier than a classic bread recipe made with wheat flour. Resist the urge to add more flour! Instead, handle the dough with moistened hands so that it doesn't stick to them.
Bake a rustic loaf instead of using a bread pan. This dough can also be shaped into a free-form loaf. Let it rise, covered, on a lightly floured piece of parchment paper for the second rise, while preheating a baking stone or steel in the oven. Slide the loaf with the paper onto a baking steel or baking sheet and bake for 35 to 40 minutes, or until the internal temperature of the loaf is 195°F to 200°F.
Allow your no-knead rye bread to cool completely before slicing. I know waiting to slice freshly baked bread is really difficult, but the cooling time is important for the crumb to set.