Answer: Most professionally propagated roses are produced by the specialized techniques of budding or grafting, but home gardeners can propagate by rooting cuttings with a little time and effort.
There are as many methods for doing this as there are rosarians. Here is one method for rooting them: First find a healthy stem tip that's the right age and size. (A good one will snap when you try to break it off.) Take stem tip with six sets of leaves on it. Remove the two bottom sets of leaves and cut off the tip just below the second set of leaves from the top. (Now you have a short piece with just two sets of leaves on it -- keep it right side up.) Some gardeners will use a rooting hormone at this stage, but it is not strictly necessary. Now stick the stem into the soil up to the bottom leaves. Firm the soil, water it lightly and cover the cutting with a large glass jar pushed securely into the soil. Usually the moisture condensing inside the jar is sufficient to keep the cutting watered, and new shoots appear in about a month. Don't take the jar off until the cutting has enough roots to support itself. (This may take the entire growing season.)
The most important part of the process is selecting the planting spot. You will need a well prepared planting bed well amended with organic matter and a location in morning sun or partial shade as the cutting should be protected from hot afternoon sun. Alternatively, you might try sticking the cutting in a pot and enclosing the whole thing in a clear plastic bag. In this case, the pot should be kept in very bright but indirect light.
Depending on what type of rose this is, you may find that cuttings root poorly. I mention this because many of the hybrid tea roses simply have weak root systems.
Roses can also be layered, which involves bending a branch down to the ground in a "U" shape so that the bent portion is buried and the growing tip is above ground. To encourage rooting, wound the bottom side of the branch slightly where it touches the soil, cover it with a few inches of soil, weight it down with a rock and top with a generous layer of mulch. Water it occasionally during the growing season. Eventually, the branch will develop enough roots from the wounded area to support itself enough to separated from the parent plant.
If the rose is a shrub rose growing on its own roots, you may be able to simply dig up and remove a sucker, or rooted shoot, from near the base of the plant. This would be an easier method of propagating it.
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