TL;DR: Calculating the right ascension (i.e. converting from ecliptic to equatorial coordinates) requires some spherical trigonometry, but fortunately there are internet tools that will do this, such as here. The resulting RA coordinates are:
Aries 0h 0m 0s; Taurus 1h 51m 39s; Gemini 3h 51m 16s; Cancer 6h 0m 0s; Leo 8h 8m 44s; Virgo 10h 8m 21s; Libra 12h 0m 0s; Scorpio 13h 51m 39s; Sagittarius 15h 51m 16s; Capricorn 18h 0m 0s; Aquarius 20h 8m 44s; Pisces 22h 8m 21s.
Explanation and background
Since traditional (Western) astrology uses the same tropical year as astronomical celestial coordinates, the tropical zodiac commences with Aries at 0˚ longitude on the ecliptic, and each subsequent zodiac sign (NB not the constellation, which is different) commences exactly 30˚ further along the ecliptic. So, the zodiac division of Taurus is from 30.00˚ to 59.99˚, Gemini from 60.00˚ to 89.99˚, and so on. It's then just a matter of converting ecliptic longitude to celestial coordinates (right ascension).
Further useful information on the relationship between the ecliptic and right ascension can be found in Mike G's answer to "Effect of the obliquity of the ecliptic / tilt of the Earth on the equation of time".
Both the traditional tropical zodiac and modern astronomical coordinates are based on the northern hemisphere vernal equinox. This starting point, also known as the First Point of Aries, is one of the two points at which the celestial equator crosses the ecliptic.
When Hipparchus defined the First Point of Aries in 130 BCE, it actually aligned with the border between the constellations of Pisces and Aries. However, due to the precession of the equinoxes, the tropical zodiac's starting point has a retrograde movement along the ecliptic at the rate of one degree every 71.5 years. As a result, the First Point of Aries is currently deep in Pisces! Using the current official IAU constellation boundaries, the point of 0h RA will cross into Aquarius in the year 2597, so we have quite some time to wait for the "Age of Aquarius".
Note that not all astrological traditions are based on the tropical year. Hindu astrology, for instance, is based on the sidereal zodiac - i.e. it aligns permanently with the visible stars, regardless of the Earth's axial precession. Some modern Western astrologers also use a sidereal zodiac, in which sidereal Aries currently begins on 15 April.